Analysis and Design of Quadrature Oscillators (Analog Circuits and Signal Processing)

ANALYSIS AND DESIGN OF QUADRATURE OSCILLATORS

ANALOG CIRCUITS AND SIGNAL PROCESSING SERIES Consulting Editor: Mohammed Ismail. Ohio State University Titles in Series: SUBSTRATE NOISE COUPLING IN RFICs Helmy, Ahmed, Ismail, Mohammed ISBN: 978-1-4020-8165-1 BROADBAND OPTO-ELECTRICAL RECEIVERS IN STANDARD CMOS Hermans, Carolien, Steyaert, Michiel ISBN: 978-1-4020-6221-6 ULTRA LOW POWER CAPACITIVE SENSOR INTERFACES Bracke, Wouter, Puers, Robert, Van Hoof, Chris ISBN: 978-1-4020-6231-5 LOW-FREQUENCY NOISE IN ADVANCED MOS DEVICES ¨ Haartman, Martin v., Ostling, Mikael ISBN-10: 1-4020-5909-4 CMOS SINGLE CHIP FAST FREQUENCY HOPPING SYNTHESIZERS FOR WIRELESS MULTI-GIGAHERTZ APPLICATIONS Bourdi, Taoufik, Kale, Izzet ISBN: 978-14020-5927-8 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 FULL-CHIP NANOMETER ROUTING TECHNIQUES Ho, Tsung-Yi, Chang, Yao-Wen, Chen, Sao-Jie ISBN: 978-1-4020-6194-3 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 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 R´ıo, R., Medeiro, F., P´erez-Verd´u, B., de la Rosa, J.M., Rodr´ıguez-V´azquez, 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 AD 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

Analysis and Design of Quadrature Oscillators by

Luis B. Oliveira Universidade Nova de Lisboa and INESC-ID, Lisbon, Portugal

Jorge R. Fernandes Technical University of Lisbon and INESC-ID, Lisbon, Portugal

Igor M. Filanovsky University of Alberta, Canada

Chris J.M. Verhoeven Technical University of Delft, The Netherlands

and

Manuel M. Silva Technical University of Lisbon and INESC-ID, Lisbon, Portugal

123

Dr. Luis B. Oliveira INESC-ID Rua Alves Redol 9 1000-029 Lisbon Portugal [email protected]

Dr. Chris J.M. Verhoeven Delft University of Technology Mekelweg 4 2628 CD Delft Netherlands [email protected]

Dr. Jorge R. Fernandes INESC-ID Rua Alves Redol 9 1000-029 Lisbon Portugal [email protected]

Dr. Manuel M. Silva INESC-ID Rua Alves Redol 9 1000-029 Lisbon Portugal [email protected]

Dr. Igor M. Filanovsky University of Alberta Dept. Electrical & Computer Engineering 87 Avenue & 114 Street Edmonton AB T6G 2V4 2nd Floor ECERF Canada [email protected]

ISBN: 978-1-4020-8515-4

e-ISBN: 978-1-4020-8516-1

Library of Congress Control Number: 2008928025 c 2008 Springer Science+Business Media B.V.  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.

Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com

To the authors’ families

Preface

Modern RF receivers and transmitters require quadrature oscillators with accurate quadrature and low phase-noise. Existing literature is dedicated mainly to single oscillators, and is strongly biased towards LC oscillators. This book is devoted to quadrature oscillators and presents a detailed comparative study of LC and RC oscillators, both at architectural and at circuit levels. It is shown that in cross-coupled RC oscillators both the quadrature error and phase-noise are reduced, whereas in LC oscillators the coupling decreases the quadrature error, but increases the phase-noise. Thus, quadrature RC oscillators can be a practical alternative to LC oscillators, especially when area and cost are to be minimized. The main topics of the book are: cross-coupled LC quasi-sinusoidal oscillators, cross-coupled RC relaxation oscillators, a quadrature RC oscillator-mixer, and twointegrator oscillators. The effect of mismatches on the phase-error and the phasenoise are thoroughly investigated. The book includes many experimental results, obtained from different integrated circuit prototypes, in the GHz range. A structured design approach is followed: a technology independent study, with ideal blocks, is performed initially, and then the circuit level design is addressed. This book can be used in advanced courses on RF circuit design. In addition to post-graduate students and lecturers, this book will be of interest to design engineers and researchers in this area. The book originated from the PhD work of the first author. This work was the continuation of previous research work by the authors from TUDelft and University of Alberta, and involved the collaboration of 5 persons in three different institutions. The work was done mainly at INESC-ID (a research institute associated with Technical University of Lisbon), but part of the PhD work was done at TUDelft and at the University of Alberta. This has influenced the work, by combining different views and backgrounds. This book includes many original research results that have been presented at international conferences (ISCAS 2003, 2004, 2005, 2006, 2007 among others) and published in the IEEE Transactions on Circuits and Systems.

vii

viii

Lisbon, Portugal Lisbon, Portugal Canada The Netherlands Lisbon, Portugal

Preface

Luis B. Oliveira Jorge R.Fernandes Igor M. Filanovsky Chris J.M. Verhoeven Manuel M. Silva

Acknowledgements

The work reported in this book benefited from contributions from many persons and was supported by different institutions. The authors would like to thank all colleagues at INESC-ID Lisboa, Delft University of Technology, and University of Alberta, particularly Chris van den Bos and Ahmed Allam, for their contributions to the work presented in this book and for their friendly and always helpful cooperation. The authors acknowledge the support given by the following institutions:

r

Fundac¸a˜ o para a Ciˆencia e Tecnologia of Minist´erio da Ciˆencia, Tecnologia e Ensino Superior, Portugal, for granting the Ph.D. scholarship BD 10539/2002, for funding projects OSMIX (POCTI/38533/ESSE/2001), SECA (POCT1/ESE/47061/2002), LEADER (PTDC/EEA-ELC/69791/2006), SPEED (PTDC/EEA-ELC/66857/2006),

and for financial support to the participation in a number of conferences.

r r r r

INESC-ID Lisboa (Instituto de Engenharia de Sistemas e Computadores – Investigac¸a˜ o e Desenvolvimento em Lisboa), Delft University of Technology, and University of Alberta, for providing access to their integrated circuit design and laboratory facilities. European Union, through project CHAMELEON-RF (FP6/2004/IST/4-027378). NSERC Canada for continuous grant support. CMC Canada for arranging integrated circuits manufacturing.

ix

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Organization of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Main Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 4 5

2 Transceiver Architectures and RF Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Receiver Architectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Heterodyne or IF Receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Homodyne or Zero-IF Receivers . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Low-IF Receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Transmitter Architectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Heterodyne Transmitters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Direct Upconversion Transmitters . . . . . . . . . . . . . . . . . . . . . . . 2.4 Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Barkhausen Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Phase-Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Examples of Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Performance Parameters of Mixers . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Different Types of Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Quadrature Signal Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 RC-CR Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Frequency Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Havens’ Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 8 8 10 11 15 15 16 17 17 18 24 26 27 29 31 31 33 34

3 Quadrature Relaxation Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Relaxation Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 High Level Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Circuit Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Quadrature Relaxation Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 High Level Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 37 38 38 39 41 41 xi

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Contents

3.3.2 Quadrature Relaxation Oscillator without Mismatches . . . . . . 3.3.3 Quadrature Relaxation Oscillator with Mismatches . . . . . . . . . 3.3.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Phase-Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Phase-noise in a Single Relaxation Oscillator . . . . . . . . . . . . . 3.4.2 Phase-noise in Quadrature Relaxation Oscillators . . . . . . . . . . 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42 46 54 56 56 60 61

4 Quadrature Oscillator-Mixer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 High Level Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Ideal Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Effect of Mismatches and Delay . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Circuit Level Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 63 64 64 67 75 79

5 Quadrature LC-Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Single LC Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Quadrature LC Oscillator Without Mismatches . . . . . . . . . . . . . . . . . . 5.4 Quadrature LC Oscillator with Mismatches . . . . . . . . . . . . . . . . . . . . . . 5.5 Q and Phase-Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Quadrature LC Oscillator-Mixer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81 81 82 85 89 92 96 98

6 Two-Integrator Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.2 High Level Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.2.1 Non-Linear Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.2.2 Quasi-Linear Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.3 Circuit Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.4 Phase-Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.6 Two-Integrator Oscillator-Mixer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.6.1 High Level Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.6.2 Circuit Implementation and Simulations . . . . . . . . . . . . . . . . . . 114 6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 7 Measurement Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 7.2 Quadrature Relaxation Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 7.2.1 Circuit Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 7.2.2 Measurement Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.3 Quadrature LC Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 7.3.1 Circuit Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Contents

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7.3.2 Measurement Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 7.4 Quadrature Oscillator-Mixer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 7.4.1 Circuit Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 7.4.2 Measurement Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 7.5 Comparison of Quadrature LC and RC Oscillators . . . . . . . . . . . . . . . . 132 7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 8 Conclusions and Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 8.2 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 A Test-Circuits and Measurement Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 A.2 Quadrature RC and LC Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 A.2.1 Test Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 A.2.2 Measurement Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 A.3 Quadrature Relaxation Oscillator-Mixer . . . . . . . . . . . . . . . . . . . . . . . . 144 A.3.1 Test Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 A.3.2 Measurement Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

Chapter 1

Introduction

Contents 1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Organization of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Main Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 4 5

1.1 Background and Motivation The huge demand for wireless communications has led to new requirements for wireless transmitters and receivers. Compact circuits, with minimum area, are required to reduce the equipment size and cost. Thus, we need a very high degree of integration, if possible a transceiver on one chip, either without or with a reduced number of external components. In addition to area and cost, it is very important to reduce the voltage supply and the power consumption [1, 2]. Digital signal processing techniques have a deep impact on wireless applications. Digital signal processing together with digital data transmission allows the use of highly sophisticated modulation techniques, complex demodulation algorithms, error detection and correction, and data encryption, leading to a large improvement in the communication quality. Since digital signals are easier to process than analogue signals, a strong effort is being made to minimize the analogue part of the transceivers by moving as many blocks as possible to the digital domain. The analogue front-end of a modern wireless communication system is responsible for the interface between the antenna and the digital part. The analogue frontend of a receiver is critical, the specifications of its blocks being more stringent than those of the transmitter. There are two basic receiver front-end architectures: heterodyne, with one intermediate frequency (IF), or more than one; homodyne, without intermediate frequency. So far, the heterodyne approach is dominant, but the homodyne approach, after remaining a long time in the research domain, is becoming a viable alternative [1, 2]. The main drawback of heterodyne receivers is that both the wanted signal and the disturbances in the image frequency band are downconverted to the IF. Heterodyne receivers have better performance than homodyne receivers when high quality RF L.B. Oliveira et al., Analysis and Design of Quadrature Oscillators,  C Springer Science+Business Media B.V. 2008

1

2

1 Introduction

(radio frequency) image-reject and IF channel filters can be used. However, such filters can only be implemented off chip (so far), and they are expensive. A high IF is required because, with a low-IF, the image frequency band is so close to the desired frequency that an image-reject RF filter is not feasible. Homodyne receivers do not suffer from the image problem, because the RF signal is directly translated to the baseband (BB), without any IF. Thus, the main drawbacks of the heterodyne approach (image interference and the use of external filters) are overcome, allowing a highly integrated, low area, low power, and low cost receiver. However, homodyne receivers are very sensitive to parasitic baseband disturbances and to 1/f noise. Quadrature errors introduce cross-talk between the I (in-phase) and Q (quadrature) components of the received signal, which in combination with additive noise increases the bit error rate (BER). A very interesting receiver approach, which combines the best features of the homodyne and of the heterodyne receivers, is the low IF receiver [3–7]. This is basically a heterodyne receiver using special mixing circuits that cancel the image frequency. Since image reject filters are not required, there is the possibility of using a low IF, allowing the integration of the whole system on a single chip [4]. The low-IF receiver relaxes the IF channel select filter specifications, because it works at a relatively low frequency and can be integrated on-chip, sometimes digitally. The image rejection is dependent on the quality of image-reject mixing, which depends on component matching and LO (local oscillator) quadrature accuracy. Thus, very accurate quadrature oscillators are essential for low-IF receivers. Conventional heterodyne structures, with high IF, make the analogue to digital converter (A/D) specifications very difficult to fulfil with reasonable power consumption; therefore, the conversion to baseband has to be done in the analogue domain. In the low-IF architecture, the two down converted signals are digitized and mixed digitally to obtain the baseband as shown in Fig. 1.1.

Analogue

Digital

LPF

A/D

LPF

A/D

BBI

LNA BBQ

I

Q I LO1

LNA - Low noise amplifier LPF - Low pass filter

Fig. 1.1 Low-IF receiver (simplified block diagram)

Q LO2

1.1 Background and Motivation

3

The LO is a key element in the design of RF frontends. The oscillator should be fully integrated, tunable, and able to provide two quadrature output signals [8–11], I and Q. In addition to frequency and phase stability, quadrature accuracy is a very important requirement of quadrature oscillators. The most often used circuits to obtain two signals in quadrature have open-loop structures, in which the errors are propagated directly to the output. Examples of such structures are [1]: 1. Passive circuits to produce the phase-shift (RC-CR network), in which the phase difference and gain are frequency dependent. 2. Oscillators working at double of the required frequency, followed by a divider by two; this method leads to high power consumption, and reduces the maximum achievable frequency. 3. An integrator with the in-phase signal at the input, followed by a comparator to obtain the signal in quadrature (aligned with the zero crossings of the integrator output); this has the disadvantage that the two signal paths are different. In recent years, significant effort has been invested in the study of oscillators with accurate quadrature outputs [9–11]. Relaxation and LC oscillators, when crosscoupled (using feedback structures), are able to provide quadrature outputs. In this book these oscillators are studied in depth, in order to understand their key parameters, such as phase-noise and quadrature error. The relaxation oscillator has been somewhat neglected with respect to the LC oscillator, as it is widely considered as a lower performance oscillator in terms of phase-noise. Although this is true for a single oscillator, it is not for cross-coupled oscillators. In this work we consider alternatives to the LC oscillator and investigate their advantages and limitations. We study in detail the quadrature relaxation oscillators in terms of their key parameters, showing that due to the cross-coupling it is possible to reduce the oscillator phase-noise and make the effect of mismatches a second order effect, thus improving the accuracy of the quadrature relationship. We show that, although stand-alone LC oscillators have a very good phase-noise performance, this is degraded when there is cross coupling. In addition to these two types of quadrature oscillators, we investigate a third type of oscillator: the two-integrator oscillator. While in the previous cases we had two oscillators with coupling to provide quadrature outputs, this oscillator is able to provide inherent quadrature outputs, with phase-noise comparable to that of a relaxation oscillator. The main advantage of this oscillator is its wide tuning range, which in a practical implementation (in the GHz range) can be about one decade. Mixers are responsible for frequency translation, upconversion and downconversion, with a direct influence on the global performance of the transceiver [1,2]. They have been realized as independent circuits from the oscillators, either in heterodyne or homodyne structures. The evolution of mixer circuits has been, so far, essentially due to technological advancements in the semiconductor industry. Here, we show that it is possible to integrate the mixing function with the quadrature oscillators. This approach has the advantage of saving area and power, leading to a more

4

1 Introduction

accurate output quadrature than that obtained with separate quadrature oscillators and mixers. We study the influence of the mixing function on the oscillator performance, and we confirm by measurement the oscillator-mixer concept. However, the main emphasis of this book is on the oscillators: the inclusion of the mixing function still requires further study. In this work we study in detail the three types of quadrature oscillators referred above, and we evaluate their relative advantages and disadvantages. Simulation and experimental results are provided which confirm the theoretical analysis.

1.2 Organization of the Book This book is organized in 8 Chapters. Following this introduction, we present a survey, in Chapter 2, of RF front-ends and their main blocks: we describe the basic receiver and transmitter architectures, then we focus on basic aspects of oscillators and mixers, and, finally, we review conventional techniques to generate quadrature signals. In Chapter 3 we present a study of the quadrature relaxation oscillator, in which we consider their key parameters: oscillation frequency, signal amplitude, quadrature relationship, and phase-noise. We use a structured approach, starting by considering the oscillator at a high level, using ideal blocks, and then we proceed to the analysis at circuit level. We present simulation results to confirm the theoretical analysis. In Chapter 4 we analyse the quadrature relaxation oscillator-mixer. We first evaluate the circuit at a high level (structured approach), deriving equations for the oscillation frequency and quadrature error of the oscillator-mixer. We show that we can inject the modulating signal in the circuit feedback loop, and we explain where and how the RF signal should be injected, to preserve the quadrature relationship. Simulation results are provided to validate theoretical results. In Chapter 5 the quadrature LC oscillator is studied in terms of the oscillation frequency, signal amplitude, Q, and phase-noise. We investigate the possibility of injecting a signal to perform the mixing function. In Chapter 6 we study the two-integrator oscillator. We proceed from a high level description to the circuit implementation, and we present simulation results. We also show the possibility of performing the mixing function in this oscillator. In Chapter 7 we present several circuit implementations to provide experimental confirmation of the theoretical results: – a 2.4 GHz quadrature relaxation oscillator and a 1 GHz quadrature LC oscillator; – two 5 GHz quadrature oscillators, one RC and the other LC, designed to be suitable for a comparative study; – a 5 GHz RC oscillator-mixer (to demonstrate the study in chapter 4). In Chapter 8 we present the conclusions and suggest future research directions. In the appendix we describe the measurement setup for the above mentioned prototypes.

1.3 Main Contributions

5

1.3 Main Contributions The work that we present in this book has led to several papers in international conferences and journals. It is believed that the main original contributions of the work are: (i) A study (in Chapter 3) of cross-coupled relaxation oscillators using a structured design approach: first with ideal blocks, and then at circuit level. Equations are derived for the oscillation frequency, amplitude, phase-noise, and quadrature relationship [12–15]. A prototype at 2.4 GHz was designed to confirm the main theoretical results (quadrature relationship and phase-noise). (ii) A study of a cross-coupled relaxation oscillator-mixer at high level (in chapter 4) [12, 16–18] and investigation of the influence of the mixing function on the oscillator performance. A 5 GHz prototype was designed to validate the oscillator-mixer concept [19]. (iii) A study of cross-coupled LC oscillators concerning Q and phase-noise (in Chapter 5) [20,21]. A comparative study of phase-noise in cross-coupled oscillators, which shows that coupled relaxation oscillators can be a good alternative to coupled LC oscillators [14]. A 1 GHz prototype confirms the increase of phase-noise in LC oscillators due to coupling [21], and two circuit prototypes at 5 GHz (RC and LC) confirm that quadrature RC oscillators might be a good alternative to quadrature LC oscillators. A minor contribution is the study of the two-integrator oscillator at high level and at circuit level (in Chapter 6), in which we show that this circuit has the advantage of a large tuning range when compared with the previous ones [22]. The work reported in this book has led to further results on quadrature oscillators, with other coupling techniques [23–25]. A pulse generator for UWB-IR based on a relaxation oscillator has been proposed recently [26]. These results, however, are outside of the scope of this book.

Chapter 2

Transceiver Architectures and RF Blocks

Contents 2.1 2.2

2.3

2.4

2.5

2.6

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Receiver Architectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Heterodyne or IF Receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Homodyne or Zero-IF Receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Low-IF Receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transmitter Architectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Heterodyne Transmitters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Direct Upconversion Transmitters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Barkhausen Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Phase-Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Examples of Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Performance Parameters of Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Different Types of Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quadrature Signal Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 RC-CR Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Frequency Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Havens’ Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 8 8 10 11 15 15 16 17 17 18 24 26 27 29 31 31 33 34

2.1 Introduction In this chapter we review the basic transceiver (transmitter and receiver) architectures, and some important front-end blocks, namely oscillators and mixers. We give special attention to the conventional methods to generate quadrature signals. We start by describing the advantages and disadvantages of several receiver and transmitter architectures. Receivers are used to perform low-noise amplification, downconversion, and demodulation, while transmitters perform modulation, upconversion, and power amplification. Receiver and transmitter architectures can be divided into two types: heterodyne, which uses one or more IFs (intermediate frequencies), and homodyne, without IF. Nowadays research is more active concerning the receiver path, since requirements such as integrability, interference rejection, and L.B. Oliveira et al., Analysis and Design of Quadrature Oscillators,  C Springer Science+Business Media B.V. 2008

7

8

2 Transceiver Architectures and RF Blocks

band selectivity are more demanding in receivers than in transmitters. The importance of accurate quadrature signals to realize integrated receivers is emphasized in this chapter. At block level, the basic aspects of oscillators are reviewed, with special emphasis on the phase-noise and its importance in telecommunication systems. The oscillators can be divided into two main groups, according to whether they have strong non-linear or quasi-linear behaviour. We present an example of each: the RC relaxation oscillator (non-linear) and the LC oscillator (linear). We survey the main characteristics of mixers, which, being responsible for frequency translation (upconversion or downconversion), are essential blocks of RF transceivers. This chapter ends by discussing the conventional methods to generate quadrature outputs, all of which employ open-loop structures. We describe in detail the most widely known method, the RC-CR network, and we also discuss two other approaches to generate quadrature outputs: frequency division and the Havens’ technique.

2.2 Receiver Architectures Receivers can be divided into three main groups: – Conventional heterodyne or IF receivers – that use one intermediate frequency (IF) or more than one intermediate frequency (multi-stage IF); – Homodyne or zero-IF receivers – that convert directly the signal to the baseband. – Low-IF receiver – this is a special case of heterodyne receiver that has become important in recent years [4], since it combines some of the advantages of the homodyne and conventional IF architectures.

2.2.1 Heterodyne or IF Receivers The heterodyne receiver was called by Armstrong as superheterodyne (patented in 1917), because the designation heterodyne had already been applied in a different context (in the area of rotating machines) [2]. This is the reason why the designation superheterodyne, instead of heterodyne, became prevalent until recently. The heterodyne receiver has been, for a long time (more than 70 years), the most commonly used receiver topology. In this approach the desired signal is downconverted from its carrier frequency to an intermediate frequency (single IF); in some cases, it is further downconverted (multi IF). The schematic of a modern IF receiver for quadrature IQ (in-phase and quadrature) signals is represented in Fig. 2.1. This receiver can be built with different technologies, GaAs, bipolar, or CMOS, and uses several discrete component filters. These filters must be implemented off-chip, with discrete components, to achieve high Q, which is difficult or impossible to obtain with integrated components. Using these high Q components, the heterodyne

2.2 Receiver Architectures

9

LPF

RF BPF

LNA

Image Reject

Channel Select

IR BPF

CS BPF

A/D

LO2

IF

DSP –90º

LPF

LO1

A/D

Fig. 2.1 Heterodyne receiver

receiver achieves high performance with respect to selectivity and sensitivity, when compared with other receiver approaches [1, 4]. This receiver can handle modern modulation schemes, which require the separation of I and Q signals to fully recover the information (for example, quadrature amplitude modulation); accurate quadrature outputs are necessary (for conversion to the baseband). The main drawback of this receiver is that two input frequencies can produce the same IF. For example, let us consider that the IF is 50 MHz and we want to downconvert a signal at 850 MHz. If we consider a local oscillator with 900 MHz, a signal at 950 MHz will be also downconverted by the mixer to the IF. This unwanted signal is called image. To overcome this problem in conventional heterodyne receivers an image reject filter is placed before the mixer as illustrated in Fig. 2.2. An important issue in heterodyne receivers is the choice of IF. With a high IF it is easier to design the image reject filter and suppress the image. However, in addition to the image, we also need to take into account interferers. At the IF frequency we must remove interferers (which are also downconverted to the IF) using a channel select filter (Fig. 2.1). Using a low IF reduces the demand on the channel select filter. Furthermore, a low IF relaxes the requirements on IF amplifiers, and makes the A/D specifications easier to fulfil. Thus, there is a trade-off in the heterodyne receiver: with high IF image rejection is easier, whereas with low IF the suppression of interferers is easier. The heterodyne architecture described above requires the use of external components. It is not a good solution for low-cost, low area, and ultra compact modern applications. The challenge nowadays is to obtain a fully integrated receiver, on a Image Reject BPF Channel

Channel

Image Image (rejected by filter)

ωLO

ω1

ωIF

Fig. 2.2 Image rejection

ωIM

ω

0

ωIF

ω

10

2 Transceiver Architectures and RF Blocks

single chip. This requires either direct conversion to the baseband, or the development of new techniques to reject the image without the use of external filters. These two possible approaches will be described next.

2.2.2 Homodyne or Zero-IF Receivers In homodyne receivers the RF spectrum is translated to the baseband in a single downconversion (the IF is zero). These receivers, also called “direct-conversion” or “zero-IF”, are the most natural solution to detect information associated with a carrier in just one conversion stage. The resulting baseband signal is then filtered with a low-pass filter, which can be integrated, to select the desired channel [1, 4]. Since the signal and its image are separated by twice the IF, this zero IF approach implies that the desired channel is its own image. Thus, the homodyne receiver does not require image rejection. All processing is performed at the baseband, and we have the more relaxed possible requirements for filters and A/Ds. Using modern modulation schemes, the signal has information in the phase and amplitude, and the downconversion requires accurate quadrature signals. The block diagram of a homodyne receiver is shown in Fig. 2.3. The filter before the LNA is optional [27], but it is often used to suppress the noise and interference outside the receiver band. This simple approach permits a highly integrated, low area, low power, and low-cost realization. Direct conversion receivers have several disadvantages with respect to heterodyne receivers, which do not allow the use of this architecture in more demanding applications. These disadvantages are related to flicker noise, channel selection, LO (local oscillator) leakage, quadrature errors, DC offsets, and intermodulation: (a) Ficker noise – The flicker noise from any active device has a spectrum close to DC. This noise can corrupt substantially the low frequency baseband signals, which is a severe problem in MOS implementation (1/f corner is about 200 kHz).

LPF

RF BPF

A/D

LO

LNA

DSP –90º

LPF

Fig. 2.3 Homodyne receiver

A/D

2.2 Receiver Architectures

11

(b) Channel selection – At the baseband the desired signal must be filtered, amplified, and converted to the digital domain. The low-pass filter must suppress the out-of-channel interferers. The filter is difficult to implement, since it must have low-noise and high linearity. (c) LO leakage – LO signal coupled to the antenna will be radiated, and it will interfere with other receivers using the same wireless standard. In order to minimize this effect, it is important to use differential LO and mixer outputs to cancel common mode components. (d) Quadrature error – Quadrature error and mismatches between the amplitudes of the I and Q signals corrupt the downconverted signal constellation (e.g., in QAM). This is the most critical aspect of direct-conversion receivers, because modern wireless applications have different information in I and Q signals, and it is difficult to implement accurate high frequency blocks with very accurate quadrature relationship. (e) DC offsets – Since the downconverted band extends down to zero frequency, any offset voltage can corrupt the signal and saturate the receiver’s baseband output stages. Hence, DC offset removal or cancellation is required in direct-conversion receivers. (f) Intermodulation – Even order distortion produces a DC offset, which is signal dependent. Thus, these receivers must have a very high IIP2 (input second harmonic intercept point) The direct conversion approach requires very linear LNAs and mixers, high frequency local oscillators with precise quadrature, and use of a method for achieving submicrovolt offset and 1/f noise. All these requirements are difficult to fulfill simultaneously.

2.2.3 Low-IF Receivers Heterodyne receivers have important limitations due to the use of external image reject filters. Homodyne receivers have some drawbacks because the signal is translated directly to the baseband. Thus, there is interest in the development of new techniques to reject the image without using filters. An architecture that combines the advantages of both the IF and the zero-IF receivers is the low-IF architecture. The low-IF receiver is a heterodyne receiver that uses special mixing circuits that cancel the image frequency. A high quality image reject filter is not necessary anymore, while the disadvantages of the zero-IF receiver are avoided. Since image reject filters are not required, it is possible to use a low IF, allowing the integration of the whole system on a single chip. The low IF relaxes the IF channel select filter specifications, and, since it works at a relatively low frequency, it can be integrated on-chip, sometimes digitally. In a low-IF receiver the value of IF is once to twice the bandwidth of the wanted signal. For example, an IF frequency

12

2 Transceiver Architectures and RF Blocks

of few hundred kHz can be used in GSM applications (200 kHz channel bandwidth), as described in [4]. Quadrature carriers are necessary in modern modulation schemes, and in low IF receivers they have an additional use: accurate quadrature signals are essential to remove the image signal. This removal depends strongly on component matching and LO (local oscillator) quadrature accuracy. Two image reject mixing techniques can be used, which have been proposed by Hartley and by Weaver. The Hartley architecture [28] has the block diagram represented in Fig. 2.4. The RF signal is first mixed with the quadrature outputs of the local oscillator. After low-pass filtering of both mixers’ outputs, one of the resulting signals is shifted by 90◦ , and a subtraction is performed, as shown in Fig. 2.4. In order to show how the image is canceled we must consider the signals at points 1, 2, and 3 (Fig. 2.4). We assume that xRF (t) = VRF cos(␻RF t) + VIM cos(␻IM t)

(2.1)

where VIM and VRF are, respectively, the amplitude of image and RF signals, and ␻IM is the image frequency. It follows that x1 (t) =

VRF VIM sin[(␻LO − ␻RF )t] + sin[(␻LO − ␻IM )t] 2 2

(2.2)

x2 (t) =

VRF VIM cos[(␻LO − ␻RF )t] + cos[(␻LO − ␻IM )t] 2 2

(2.3)

Equation (2.2) can be written as: x1 (t) = −

VRF VIM sin[(␻RF − ␻LO )t] + sin[(␻LO − ␻IM )t] 2 2 sin (ω LO t) 1 90°

LPF

LO

RF Input

IF Output

–90°

LPF cos (ω LO t)

Fig. 2.4 Hartley architecture (single output)

3

2

(2.4)

2.2 Receiver Architectures

13

Since a shift of 90◦ is equivalent to a change from cos(␻t) to sin(␻t): x3 (t) =

VRF VIM cos[(␻RF − ␻LO )t] − cos[(␻LO − ␻IM )t] 2 2

(2.5)

Adding (2.5) and (2.3) cancels the image band and yields the desired signal. In Hartley’s approach, the quadrature downconversion followed by a 90◦ phase shift produces in the two paths the same polarities for the desired signal, and opposite polarities for image. The main drawback of this architecture is that the receiver is very sensitive to the local oscillator quadrature errors and to mismatches in the two signal paths, which cause incomplete image cancellation. The relationship between the image average power (PIM ) and the signal average power (PS ) is [1]: 2 V 2 (VLO + ⌬VLO )2 − 2VLO (VLO + ⌬VLO ) cos(␪) + VLO PIM = IM 2 2 PS VRF (VLO + ⌬VLO )2 + 2VLO (VLO + ⌬VLO ) cos(␪) + VLO

(2.6)

where VLO is the amplitude of local oscillator, ⌬VLO is the amplitude mismatch, and ␪ is the quadrature error. Noting that VIM 2 /VRF 2 is the image-to-signal ratio at the receiver input (RF), the image rejection ratio (IRR) is defined as PIM /PS at the IF output divided by VIM 2 /VRF 2 .  PIM  PS out IRR =  2  VIM  V2 

(2.7)

RF in

The resulting equation can be simplified if the mismatch is small (⌬VLO  VLO ) and the quadrature error ␪ is small [1, 2]:  IRR ≈

 ⌬VLO 2 + ␪2 VLO 4

(2.8)

Note that we have considered only errors of the amplitude and phase in the local oscillator. Mismatches in mixers, filters, adders, and phase shifter will also contribute to the IRR. In integrated circuits, without using calibration techniques, the typical values for amplitude mismatch are 0.2–0.6 dB and for the quadrature error 3−5◦ , leading to an image suppression of 25 to 35 dB [1, 2, 28, 29]. The second type of image-reject mixing is performed by the Weaver architecture [30], represented in Fig. 2.5. This is similar to the Harley architecture, but the 90◦ phase shift in one of the signal paths is replaced by a second mixing operation in both signal paths: the second stage of I and Q mixing has the same effect of the 90◦ phase shift used in the Hartley approach. As with the Hartley receiver, if the

14

2 Transceiver Architectures and RF Blocks

LPF

RF Input

LO1

IF or BB Output

LO2

–90°

–90°

LPF

Fig. 2.5 Weaver architecture with single output

phase difference of the two local oscillator signals is not exactly 90◦ , the image is no longer completely cancelled. The Weaver architecture has the advantage that the RC-CR mismatch effect (this effect will be discussed in detail in Section 2.6) on the 90◦ phase shift after the downconversion in the Hartley architecture is avoided and the second order distortion in Channel Secondary Image

RE Input

ωLO1

2ωLO2 – ωIN + 2ωLO1

ωIN

ω

Channel Secondary Image

First IF

2ωLO2 – ωIN + ωLO1 ωLO2

ωIN – ωLO1

ω

Channel

Second IF

Secondary Image

ωIN – ωLO1 – ωLO2

Fig. 2.6 Secondary image problem in the Weaver architecture

ω

2.3 Transmitter Architectures

15

LPF

LPF

BB I

RF Input LPF

LPF

I

BB Q

Q I LO1

Q LO2

Fig. 2.7 Weaver architecture with quadrature outputs

the signal path can be removed by the filters following the first mixing. However, as the Hartley architecture, the Weaver architecture is sensitive to mismatches in amplitude and quadrature error of the two LO signals. It suffers from an image problem (in the second mixing operation) if the second downconversion is not to the baseband, as shown in Fig. 2.6. In this case the low pass filters, must be replaced by bandpass filters, to suppress the secondary image, but, the image suppression is easier at IF than at RF. The receiver of Fig. 2.5 needs to be modified to provide baseband quadrature outputs, which are necessary in modern wireless applications. We need 6 mixers to cancel the image and separate the quadrature signals, as shown in Fig. 2.7. The second two mixers in Fig. 2.5 are replaced by two pairs of quadrature mixers, and their outputs are then properly combined [2]. This modified Weaver architecture is used in low-IF receivers [31].

2.3 Transmitter Architectures Transmitter architectures can be divided into two main groups: – Heterodyne – that use an intermediate frequency; – Direct upconversion – that converts directly the signal to the RF band.

2.3.1 Heterodyne Transmitters The heterodyne upconversion, represented in Fig. 2.8, is the most often used architecture in transmitters. In heterodyne transmitters the baseband signals are modulated in quadrature (modern transmitters must handle quadrature signals) to the IF, since it is easier to provide accurate quadrature outputs at IF than at RF. The IF filter that follows rejects the harmonics of the IF signal, and reduces the transmitted noise.

16

2 Transceiver Architectures and RF Blocks

D/A

IF BPF

LO

DSP

RF BPF

PA

–90°

LO

D/A

Fig. 2.8 Heterodyne transmitter

The IF modulated signal is then upconverted, amplified (by the power amplifier), and transmitted by the antenna. A heterodyne transmitter requires an RF band-pass filter to suppress (50–60 dB) the unwanted sideband after the upconversion, in order to meet spurious emission levels imposed by the standards. This filter is typically passive and built with expensive off-chip components [1, 4]. This topology does not allow full integration of the transmitter, due to the off-chip passive components in IF and RF filters.

2.3.2 Direct Upconversion Transmitters In this type of transmitter, shown in Fig. 2.9, the baseband signal is directly upconverted to RF. The RF carrier frequency is equal to the LO frequency, at the mixers input. A quadrature upconversion is required by modern modulations schemes. This topology can be easily integrated, because there is no need to suppress any mirror signal generated during the upconversion. As in the receiver, the local oscillator frequency is the carrier frequency [4].

D/A

LO

DSP –90°

D/A

Fig. 2.9 Direct upconversion transmitter

PA

HF BPF

2.4 Oscillators

17

The main disadvantage is the “injection pulling” or “injection locking” of the local oscillator by the high level PA output. The resulting spectrum can not be suppressed by a bandpass filter, because it has the same frequency as the wanted signal. To avoid this effect the isolation required is normally higher than 60 dB. As in the receiver case, a solution that tries to combine the advantages of both direct and heterodyne upconversion was proposed in [32, 33]. In this case the baseband signals are converted to a low IF, and are then upconverted to the final carrier frequency using an image-reject mixing technique to reject the unwanted sideband, thus avoiding the use of an RF filter after the upconversion. Thus, an integrated circuit realization is possible, with lower area and cost than with a conventional heterodyne approach.

2.4 Oscillators 2.4.1 Barkhausen Criterion The basic function of an oscillator is to convert DC power into a periodic signal. A sinusoidal oscillator generates a sinusoid with frequency ␻0 and amplitude V0 (Fig. 2.10), vOUT (t) = V0 cos(␻0 t + ␪)

(2.9)

For digital applications, oscillators generate a clock signal, which is a squarewaveform with period T0 . Sinusoidal oscillators can be analyzed as a feedback system, shown in Fig. 2.11, with the transfer function. H ( j␻) Yout ( j␻) = X in ( j␻) 1 − H ( j␻)␤( j␻)

Amplitude [V]

(2.10)

P[dBm]

V0

t[s]

(a)

ω0

(b)

Fig. 2.10 Sinusoidal oscillator output: (a) Time domain; (b) Frequency domain

ω[rad/s]

18

2 Transceiver Architectures and RF Blocks

Fig. 2.11 Feedback system block diagram

β( jω) Xin ( jω) +

Yout ( jω)

+

H( jω)

+

The necessary conditions concerning the loop gain for steady-state oscillation with frequency ␻0 are known as the Barkhausen conditions. The loop gain must be unity (gain condition), and the open-loop phase shift must be 2k␲, where k is an integer including zero (phase condition). |H ( j␻0 )␤( j␻0 )| = 1

(2.11)

arg[H ( j␻0 )␤( j␻0 )] = 2k␲

(2.12)

The Barkhausen criterion gives the necessary conditions for stable oscillations, but not for start-up. For the oscillation to start, triggered by noise, when the system is switched on, the loop gain must be larger than one, |H ( j␻)␤( j␻)| > 1 [34].

2.4.2 Phase-Noise 2.4.2.1 Definition In modern transceiver applications the most important difference between ideal and real oscillators is the phase-noise. The noise generated at the oscillator output causes random fluctuation of the output amplitude and phase. This means that the output spectrum has bands around ␻0 and its harmonics (Fig. 2.12). With the increasing order of the harmonics of ␻0 the power in the sidebands decreases [35]. The noise can be generated either inside the circuit (due to active and passive devices) or outside (e.g., power supply). Effects such as nonlinearity and periodic

P [dBm]

Fig. 2.12 Spectrum of oscillator output with phase-noise

White noise floor

ω0

2ω0

3ω0

ω[rad/s]

2.4 Oscillators

19

variation of circuit parameters make it very difficult to predict phase-noise [36]. The noise causes fluctuations of both amplitude and phase. Since, in practical oscillators there is an amplitude stabilization scheme, which attenuates amplitude variations, phase-noise is usually dominant. The oscillator noise can be characterized either in the frequency domain (phase-noise), or in the time domain (jitter). The first is used by analog and RF designers, and the second is used by digital designers [35–37]. There are several ways to quantify the fluctuations of phase and amplitude in oscillators (a review of different standards and measurement methods is presented in [38]). They are often characterized in terms of the single sideband noise spectral density, L(␻), expressed in decibels below the carrier per hertz (dBc/Hz). This characterization is valid for all types of oscillators and is defined as: L(␻m ) =

P(␻m ) P(␻0 )

(2.13)

where: P(␻m ) is the single sideband noise power at a distance of ␻m from the carrier (␻0 ) in a 1 Hz bandwidth; P(␻0 ) is the carrier power. The advantage of this parameter is its ease of measurement. This can be done by using a spectrum analyzer (which is a general-purpose equipment, but will introduce some errors) or with phase or frequency demodulators with well known properties (which are dedicated and expensive equipment). Note that the spectral density (2.13) includes both phase and amplitude noise, and they can not be separated. However, practical oscillators have an amplitude stabilization mechanism, which strongly reduces the amplitude noise, while the phase-noise is unaffected. Thus, equation (2.13) is dominated by the phase-noise and L(␻m ) is known simply as “phase-noise”. The carrier-to-noise ratio (CNR) can also be used to specify the oscillator phasenoise. The CNR in a 1 Hz frequency band at the distance of ␻m from the carrier ␻0 , is defined as: CNR(␻m ) = 1/L(␻m )

(2.14)

2.4.2.2 Quality Factor The Quality Factor (Q) is the most common figure of merit for oscillators, and it is related to the total oscillator phase-noise. Q is usually defined within the context of second order systems. There are three possible definitions of Q [1]: (1) Leeson in [39] considers a single resonator network with −3 dB bandwidth B and resonance frequency ␻0 (Fig. 2.13),

20

2 Transceiver Architectures and RF Blocks

Fig. 2.13 Q definition for a second order system

H(s)

3 dB

ω0

ω

B

Q=

␻0 B

(2.15)

A second order bandpass filter has a transfer function: ␻0 s Q H (s) = ␻0 s2 + s + ␻20 Q K

(2.16)

where K is the mid-band gain and Q is the pole quality factor (the same Q of (2.15)). For Q  1 the transfer function is symmetric as shown in Fig. 2.13. In practice this approximation is valid for Q ≥ 5. This definition of Q is suitable for filters, and can be used in oscillators if we consider the resonator circuit as a second order filter. (2) A second definition of Q considers a generic circuit and relates the maximum energy stored and the energy dissipated in a period [2]: Q = 2␲

Maximum energy stored in a period Energy dissipated in a period

(2.17)

This definition is usually applied to a general RLC circuit and relates the maximum energy stored (in C or L) and the energy dissipated (by R) in a period. As an example, we apply the definition to an RLC series circuit. The energy is stored in the inductor and the capacitor, and the maximum energy stored in the inductor and the capacitor is the same. The energy stored in an inductor (WL ) is: T WL =

i(t)L

di(t) dt = L Ir2ms dt

(2.18)

0

where Ir ms is the root-mean-square current in the inductor. The energy dissipated in a resistor (W R ) per cycle (in the period T0 ) is: W R = Ir2ms RT0

(2.19)

2.4 Oscillators

21

The value of Q is: Q = 2␲

L ␻0 L L Ir2ms = 2␲ f 0 = Ir2ms RT0 R R

(2.20)

We can also use the energy stored in the capacitor: T WC =

ν(t)C

dν(t) dt = C Vr2ms dt

(2.21)

0

where Vr ms is the root-mean-square voltage in the capacitor, and Vr ms = Ir ms /(␻0 C). Then, the value of Q is: Q = 2␲

(C Ir2ms )/(␻20 C 2 ) 1 C Vr2ms = 2␲ f = 0 2 2 Ir ms RT0 R Ir ms ␻0 C R

(2.22)

Equation (2.17) is a general definition, which does not specify which elements store or dissipate energy. (3) In the third definition of Q the oscillator is considered as a feedback system and the phase of the open-loop transfer function H (j␻) is evaluated at the oscillation frequency, ␻osc , which is not necessarily the resonance frequency [36]. In a single RLC circuit the oscillation frequency is the resonance frequency, but with coupled oscillators the oscillation frequency can be different, as we will show in Chapter 5. The oscillator Q is defined as: ␻0 Q= 2





dA d␻



2 +

d␪ d␻

2 (2.23)

where A is the amplitude and ␪ is the phase of H (j␻). This definition, called openloop Q, was proposed in [36], and takes into account the amplitude and phase variations of the open-loop transfer function. This Q definition is often applied to a single resonator as shown in Fig. 2.14. This definition is very useful to calculate the oscillator quality factor, which has its maximum value at the resonance frequency, and it will be used in Chapter 5 to calculate the degradation of the oscillator quality factor if the oscillation frequency is θ = ∠H( jω)

H( jω) L

Fig. 2.14 Definition of Q based on open-loop phase slope

C

R

ω0

ω

22

2 Transceiver Architectures and RF Blocks

different from the resonance frequency. We will also use this definition in Chapter 6 to calculate the quality factor of a two-integrator oscillator. 2.4.2.3 Leeson-Cutler Phase-Noise Equation The most used and best known phase-noise model is the Leeson-Cutler semi empirical equation proposed in [39–41]. It is based on the assumption that the oscillator is a linear time invariant system. The following equation for L(␻m ) is obtained [42]: 

2FkT L(␻m ) = 10 log PS



 1+

␻0 2Q␻m

2 

␻1/ f 3 1+ |␻m |



(2.24)

where: k – Boltzman constant; T – absolute temperature; PS – average power dissipated in the resistive part of the tank; ␻0 – oscillation frequency; Q – quality factor (also known as loaded Q); ␻m – offset from the carrier; ␻1/ f 3 – corner frequency between 1/ f 3 and 1/ f 2 zones of the noise spectrum (represented in Fig. 2.15); F – empirical parameter, called excess noise factor. A detailed study of this parameter, which includes nonlinear effects for LC oscillators, was done in [43]. A different model to predict the oscillator phase-noise was presented recently in [42]. This is a linear time-variant model, which, according to the authors, gives accurate results without any empirical or unspecified factor. In Fig. 2.15, a typical asymptotic output noise spectrum of an oscillator is shown. This plot has three different regions [35]:

(ω) (dBc/Hz) –30 dB/decade

–20 dB/decade white noise floor

Fig. 2.15 A typical asymptotic noise spectrum at the oscillator output

(3)

ω0

(2)

ω1

(1)

ω2

ω

2.4 Oscillators

23

(1) For frequencies far away from the carrier, the noise of the oscillator is due to white-noise sources from circuits, such as buffers, which are connected to the oscillator, so there is a constant floor in the spectrum. (2) A region [␻1 −␻2 ] with a −20 dB/decade slope is due to FM of the oscillator by its white noise sources. (3) In the region close to the carrier frequency, with frequencies between ␻0 and ␻1 there is a −30 dB/decade slope due to the 1/f noise of the active devices.

2.4.2.4 Importance of Phase-Noise in Wireless Communications The phase-noise in the local oscillator will spread the power spectrum around the desired oscillation frequency. This phenomenon will limit the immunity against adjacent interferer signals: in the receiver path we want to downconvert a specific channel located at a certain distance from the oscillator frequency; due to the oscillator phase-noise, not only the desired channel is downconverted to an intermediate frequency, but also the nearby channels or interferers, corrupting the wanted signal [35] (Fig. 2.16). This effect is called “reciprocal mixing”. In the case of the transmitter path the phase-noise tail of a strong transmitter can corrupt and overwhelm close weak channels [35] (Fig. 2.17). As an example, if a receiver detects a weak signal at ␻2 , this will be affected by a close transmitter signal at ␻1 with substantial phase-noise.

Interferer

Interferer Signal

Fig. 2.16 Phase-noise effect on the receiver and the undesired downconversion

ω0

Signal

ω

ωIF

ω

Close Transmitter

Signal

Fig. 2.17 Phase-noise effect on the transmitter path

ω 1 ω2

ω

24

2 Transceiver Architectures and RF Blocks

2.4.3 Examples of Oscillators Oscillators can be divided into two main groups: quasi-linear and strongly nonlinear oscillators [34]. Strongly non-linear or relaxation oscillators are usually realized by RC-active circuits. In this book we will present a detailed study of relaxation oscillators. The main advantage of this type of oscillators is that only resistors and capacitors are used together with the active devices (inductors, which are costly elements in terms of chip area, are not needed); the main drawback of relaxation oscillators is their high phase-noise. In addition to the relaxation oscillator, another RC oscillator will be studied: the two-integrator oscillator. This oscillator is very interesting because it can have either linear or non-linear behaviour, as we will see in detail in Chapter 6. LC oscillators are usually quasi-linear oscillators. They can use as resonator element: dielectric resonators, crystals, striplines, and LC tanks. These oscillators are known by their good phase-noise performance, since Q is normally much higher than one. In this book we are interested in oscillators capable to produce two outputs in accurate quadrature. We will study relaxation RC oscillators and LC oscillators with an LC tank (usually called simply LC oscillators), because they can be cross-coupled to provide quadrature outputs. We also study a third type of oscillator: the twointegrator oscillator, which has inherent quadrature outputs. In the next part of this section we present examples of an RC relaxation oscillator and of an LC oscillator. 2.4.3.1 Relaxation Oscillators Relaxation oscillators are widely used in fully integrated circuits (because they do not have inductors), in applications with relaxed phase-noise requirements [9], typically as part of a phase-locked loop. However, these oscillators have not been popular in RF design because they have noisy active and passive devices [1]. VCC

R

R

M

M

C I

Fig. 2.18 Relaxation oscillator

I

2.4 Oscillators

25

In Fig. 2.18 we present an example of an RC relaxation oscillator. This oscillator has been referred to as a first order oscillator, since its behaviour can be described in terms of first order transients [8, 44]. It operates by alternately charging and discharging a capacitor between two threshold voltage levels that are set internally. The oscillation frequency cannot be determined by the Barkhausen criterion (this is not a linear oscillator) and it is inversely proportional to capacitance. 2.4.3.2 LC Oscillator In order to illustrate the Barkhausen criterion, the LC oscillator can be used because it is a quasi-linear oscillator. Oscillation will occur at the frequency for which the amplitude of the loop gain is one and the phase is zero. The LC oscillator model is represented in Fig. 2.19: the transfer function is H ( j␻) = gm and ␤( j␻) is the impedance of the parallel RLC circuit. 

␤( j␻) = 1+ j

R  ␻ ␻0 − Q ␻0 ␻

(2.25)

where  Q=R

␻0 = √

C L

(2.26)

1

(2.27)

LC

At the resonance frequency (␻0 ) the inductor and capacitor admittances cancel and the loop gain is |H (j␻0 )␤( j␻0 )| = gm R = 1: the active circuit has a negative resistance, which compensates the resistance of the parallel RLC circuit. This condition is necessary, but not sufficient, because, for the oscillation to start, the loop gain must be higher than 1, gm > 1/R. In Fig. 2.20 a typical LC oscillator, used in RF transceivers, is shown. This is known as LC oscillator with LC-tank, and it is also called differential CMOS LC

gm C

Fig. 2.19 LC-oscillator behavioural model

R

L

26

2 Transceiver Architectures and RF Blocks

Fig. 2.20 CMOS LC Oscillator with LC tank

VCC L

C

L

M

M

C

I

Fig. 2.21 Equivalent resistance of the differential pair

+vx–

ix vx 2

–gm

vx 2

–

–

vx 2

vx 2

vx 2

gm

vx 2

oscillator, or negative gm oscillator. The cross-coupled NMOS transistors (M) generate a negative resistance, which is in parallel with the lossy LC tank (Fig. 2.21). In Fig. 2.21 the small signal model of the differential pair is shown. Since the circuit is symmetric, the controlled sources have the currents shown in Fig. 2.21, and the equivalent resistance of the differential pair is: Rx =

vx 2 =− ix gm

(2.28)

Thus, the differential pair realizes a negative resistance (Fig. 2.21) that compensates the losses in the tank circuit.

2.5 Mixers Mixers are a fundamental block of RF front-ends. Nowadays, a research effort is done to realize a fully integrated front-end, to obtain cost and space savings. Integrated mixers are usually a separate block of the receiver; however, the possibility

2.5 Mixers

27

of combining the LNA and the mixer [45] has been considered. In this book we will investigate the combination of the oscillator and the mixer in a single block. Conventional mixers have an open-loop structure, in which the output is obtained by the multiplication of a local oscillator signal and an input signal (RF signal, in the receiver path). For quadrature modulation and demodulation two independent mixers are required, which imposes severe constrains on the matching of circuit components. The integration of the mixing function in quadrature oscillators has the advantage of relaxing these constraints, as will be shown in Chapter 4 of this book. In this section we review the most important characteristics of mixers: noise figure, second and third order intermodulation points, 1-dB compression point, gain, input and output impedance, and isolation between ports. Different types of implementations will be reviewed [1, 2].

2.5.1 Performance Parameters of Mixers The noise factor (NF) is the ratio of the signal-to-noise ratios at the input and at the output. It is an important measure of the performance of the mixer, indicating how much noise is added by it. The noise factor of a noiseless system is unity, and it is higher in real systems. NF =

SNRIN SNROUT

(2.29)

The intermodulation distortion (IMD) is a measure of the mixers linearity. Intermodulation distortion is the result of two or more signals interacting in a non linear device to produce additional unwanted signals. Two interacting signals will produce intermodulation products at the sum and difference of integer multiples of the original frequencies. For two input signals at frequencies f 1 and f 2 , the output components will have frequencies m f 1 ± n f 2 , where m and n are integers. The second and third-order intercept points (IP2 and IP3 ) can be defined for the input (IIP2 and IIP3 ), or for the output (OIP2 and OIP3 ), as represented in Fig. 2.22. Here, the desired output (P1 ) and the third order IM output (P3 ) are represented as a function of the input power. IIP3 and OIP3 are the input and output power, respectively, at the point of intersection (extrapolated) of the two lines. The IIP3 can be determined for any input power (PIN ) from the difference of power between the signal and third harmonic (⌬P) as shown in Fig. 2.22 [1]. It can be shown that there is a relationship between the IIP3 and ⌬P for a given PIN [1], as indicated in Fig. 2.22. Using the same procedure, we can obtain the IP2 , and the respective input and output intercept points (IIP2 and OIP2 ), which are obtained from the intersection point of P1 and the second order IM output power (not represented in Fig. 2.22). In a receiver with IF (heterodyne), the third-order intermodulation distortion is the most important. If two input tones at f 1 + f LO and f 2 + f LO are close in frequency, the intermodulation components at 2 f 2 − f 1 and 2 f 1 − f 2 will be close to f 1 and

28

2 Transceiver Architectures and RF Blocks POUT (dBm)

IP3

OIP3 P1

f1 f2

ΔP ΔP

2f1 − f2

2f2 − f1

P3

f

ΔP 2

IIP3 = ΔP + PIN (dBm) 2

IIP3

(a)

(b)

PIN (dBm)

Fig. 2.22 (a) Calculation of IIP3. (b) Graphical Interpretation

f 2 , making them difficult to filter without also removing the desired signal. Higher order intermodulation products are usually less important, because they have lower amplitudes, and are more widely spaced. The remaining third order products, 2 f 1 + f 2 and 2 f 2 + f 1 , do not present a problem. The second-order intermodulation distortion is important in direct conversion (homodyne receivers). In this case, intermodulation due to two input signals ( f 1 and f 2 ), can be close to DC ( f 2 − f 1 and f 1 − f 2 ), and lie in the signal band (Fig. 2.23). Thus, a mixer that converts directly to the baseband has very stringent IP2 requirements. Another specification concerning distortion is the 1-dB compression point. This is the output power when it is one dB less than the output power of an extrapolated linear amplifier with the same gain (Fig. 2.24). The conversion gain of a mixer can be defined in terms of either voltage or power. – The voltage conversion gain is defined as the ratio of rms voltage of the IF signal to the rms voltage of the RF signal.

Signal

Signal f1 − f2

fLO

f1 + fLO f2 + fLO RF

f2 − f1

0 fLO

Fig. 2.23 Second order distortion in a direct conversion mixer

Baseband

f1

f2

2.5 Mixers Fig. 2.24 Calculation of P-1dB

29

1 dB

POUT (dB)

P–1dB



VOUT Voltage Gain(dB) = 20 log VIN

PIN (dB)

 (2.30)

– The power conversion gain is defined as the IF power delivered to a load (RL ) divided by the available power from an RF source with resistance RS .  Power Gain(dB) = 10 log

POUT PIN (available)

 (2.31)

If the load impedance is equal to the source impedance (for example 50 ⍀) then the voltage and power conversion gains are equal. In conventional heterodyne receivers the input impedance of the mixer must be 50 ⍀ because we need an external image reject filter, which should be terminated by 50 ⍀ impedance. In other receiver architectures, which do not need off-chip filters (e.g., low IF receiver), there is no need for 50 ⍀ matching, but the mixer input needs to be matched to the LNA output. The isolation between the mixer ports is critical. This quantifies the interaction among the RF, IF (or baseband for homodyne receivers), and LO ports. The LO to RF feedthrough results in LO leakage to the LNA, and eventually to the antenna; the RF to LO feedthrough allows strong interferers in the RF path to interact with the local oscillator that drives the mixer. The LO to IF feedthrough is undesirable because substantial LO signal at the IF output will disturb the following stages.

2.5.2 Different Types of Mixers There are several types of possible implementations for a mixer. The choice of the implementation is based on linearity, gain, and noise figure requirements. The simplest mixer is a switch, implemented by a CMOS transistor [1]. The circuit of Fig. 2.25 is referred to as a passive mixer; although having an active element, the transistor, this acts as a switch, and does not provide gain. This type of mixers,

30

2 Transceiver Architectures and RF Blocks

Fig. 2.25 Mixer using a switch

vLO

vRF

vIF RL

typically has no DC consumption, has high linearity and high bandwidth, and is suitable for use in microwave circuits. There are other possible implementations, as shown in Figs. 2.26 and 2.27, which, by contrast, provide gain, and reduce the effect of noise generated by subsequent stages; they are referred to as active mixers. These are widely used in RF systems, and most of them are based on the differential pair. They can be divided into single-balanced mixers, where the LO frequency is present in the output spectrum and double-balanced mixers, which use symmetry to remove the LO frequency from the output. In the single-balanced mixer, the differential pair has the LO signal at the input and the current source is controlled by the other input signal (RF signal in downconversion, as shown in Fig. 2.26). It converts the RF input voltage to a current, which is steered either to one or to the other side of the differential pair. This mixer has the advantage that it is simple to design and operates with a single-ended RF input. VCC

R

R

vIF

M1

M2

vLO

vRF

Fig. 2.26 Active single-balanced mixer

M3

2.6 Quadrature Signal Generation

31 VCC

Fig. 2.27 Active double-balanced mixer

R

R

vIF

M2

M1

M3

M4

vLO

vRF

M5

M6

I

When compared with a double-balanced mixer, it has moderate gain and moderate noise figure, low 1 dB compression point, low port-to-port isolation, low IIP3 , and high input impedance (this can be an advantage if the mixer does not have a 50 ⍀ load) [46]. The double balanced mixer is a more complex circuit, which has LO and RF differential inputs: it is the Gilbert cell [1, 2], represented in Fig. 2.27. This mixer has higher gain, lower noise figure, good linearity, high port-to-port isolation, high spurious rejection, and less even order distortion, with respect to the single balanced mixer. The main disadvantage is the increased area (due to complexity) and power consumption [1,46]; additionally, it may require a balun transformer [46] to provide the RF differential input (the image reject filter output is typically single-ended).

2.6 Quadrature Signal Generation In modern transceivers, accurate quadrature is required for modulation and demodulation and for image rejection. The common methods of generating signals with a phase difference of 90◦ , employ open-loop structures [1], and are reviewed in this section. We analyse in detail the RC-CR network, which is the best known approach, and we present other techniques that can be found in the literature: frequency division and Heaven’s technique.

2.6.1 RC-CR Network This is the simplest technique and uses an RC-CR network (Fig. 2.28), in which the input is shifted by +45◦ in the CR branch and by −45◦ in the RC branch. The outputs are in quadrature at all the frequencies, but the amplitude is not constant [2]. The phase shift of vOUT1 is zero at DC and by increasing the frequency decreases asymptotically to −90◦ . The phase shift of vOUT2 is +90◦ at DC and decreases

32

2 Transceiver Architectures and RF Blocks

Fig. 2.28 Quadrature generation using an RC-CR circuit C

R

vOUT1 vIN

C

vOUT2 R

with the frequency towards 0◦ . The phase shift of each branch changes with the frequency, but the phase difference of the two outputs is always 90◦ . This approach provides a good quadrature relationship, but the amplitude of the outputs changes significantly with the frequency. The I and Q branches have, respectively, a low-pass and a high-pass characteristic. The two output amplitudes are only equal at the pole frequency, ␻ p = 1/RC. The design procedure is simply to set the pole frequency to the carrier frequency. However, the absolute value of RC varies with temperature and with process, having a direct influence on the value of the frequency at which there are quadrature signals with equal amplitude. To minimize this problem, the amplitudes can be equalized by using limiter stages based on differential pairs [1] or using variable gain amplifiers [2]. At the pole frequency there is 3 dB attenuation, which is a significant loss. Moreover, this network generates thermal noise, which can not be ignored. In the circuit of Fig. 2.28, the mismatch of resistors and capacitors originates a deviation ␪ from the 90◦ phase difference. Assuming relative mismatches ␣ for the resistances and ␤ for the capacitances, we can express ␪ in the neighbourhood of ␻ = 1/(RC) as:

␪=

␲ − {arctan[R(1 + ␣)C(1 + ␤)␻] − arctan(RC␻)} 2

(2.32)

Using the trigonometric relationship

arctan(A) − arctan(B) =

A−B 1 + AB

(2.33)

2.6 Quadrature Signal Generation

33

we obtain ␲ ␪ = − arctan 2



RC␻(1 + ␣)(1 + ␤) − (RC␻) 1 + RC␻(1 + ␣)(1 + ␤)RC␻

 (2.34)

If ␣  1 and ␤  1 (small mismatches), and taking into account that ␻ ≈ 1/(RC)):   ␣+␤ ␲ ␪ ≈ − arctan 2 2

␪≈

(2.35)

␲ ␣+␤ − 2 2

(2.36)

For typical values ␣ = ␤ = 10%, equation (2.36) gives 5.73◦ worst-case quadrature error. An RC-CR network with two or more stages is known as a polyphase filter. A single RC-CR stage provides (without mismatches) an amplitude error below 0.2 dB over a 10% bandwidth. A properly designed 2-stage RC-CR network can give the same gain error with a higher bandwidth. We can use more stages in order to cover the required bandwidth. However, a polyphase filter has significant attenuation and high noise [2]. To avoid these problems other quadrature techniques may be used, which provide inherently quadrature outputs with equal amplitude.

2.6.2 Frequency Division Another approach to generate quadrature carriers is frequency division. This is a simple technique in which a master-slave flip-flop is used to divide by two the frequency of a signal with double of the desired frequency (Fig. 2.29). If vIN has 50% duty-cycle, then the outputs are in quadrature [1]. The use of a carrier with twice the desired frequency has two main disadvantages: there is an increase in the power consumption, and the maximum achievable frequency is reduced. Mismatches in the signal paths through the latches and

Latch

vOUT1

Fig. 2.29 Frequency divider as a quadrature generator

vOUT2

vIN

Latch

34

2 Transceiver Architectures and RF Blocks

deviations of the input duty-cycle from 50% contribute to the phase error. In order to reduce the quadrature error, two dividers can be used, but this requires an input signal with 4 times the required frequency [2].

2.6.3 Havens’ Technique A third method of quadrature generation, less often used, is Havens’ technique, which is represented in (Fig. 2.30a). The input signal is split into two branches by using a phase shifter by approximately 90◦ , generating v1 and v2 : v1 = A cos(␻t)

(2.37)

v2 = A cos(␻t + ␪)

(2.38)

The soft-limiter stages are used to equalize the amplitudes of v1 and v2 after the phase shifter (RC-CR network). After this limiting action, the two signals are added and subtracted, and the results are again limited, generating the two final outputs, which are approximately sinusoidal (since the limiter is “soft”) and in quadrature:   ␪ ␪ v1 (t) + v2 (t) = 2A cos cos ␻t + 2 2

(2.39)

  ␪ ␪ sin ␻t + 2 2

(2.40)

v1 (t) − v2 (t) = 2A sin

vOUT1

Soft-limiter

vOUT1

v1 vIN

v2

v1

~90° vOUT2

v2

vOUT2 –v2

(a)

Fig. 2.30 (a) Havens quadrature generator circuit. (b) Phasor diagram

(b)

2.6 Quadrature Signal Generation

35

The main advantage of this approach is that, although any error in the 90◦ phase shift block leads to an amplitude mismatch between the two outputs, this is cancelled by the soft-limiters, as shown in equations (2.39) and (2.40). This method is robust with respect to amplitude errors; however, the need of four soft-limiters and two adders makes this circuit less attractive for low-power, low area, and low cost applications. The above analysis assumes quasi-sinusoidal signals. However, the soft-limiters and the non-linearity of the adders generate harmonics. This is an important drawback of this approach: even order harmonics with 90◦ phase difference results in quadrature errors, and odd order harmonics produce amplitude mismatch. Finally, the capacitive coupling between the inputs of the two adders is an extra source of quadrature error [1]. It is important to note that in the Havens technique the generated quadrature signals are usually quasi-sinusoidal, while in the frequency division approach the outputs have a square waveform. All the conventional quadrature generating circuits, reviewed above, have openloop architectures, in which the errors are propagated to the output. In this book, we will study closed-loop architectures, which have better quadrature accuracy.

Chapter 3

Quadrature Relaxation Oscillator

Contents 3.1 3.2

3.3

3.4

3.5

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relaxation Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 High Level Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Circuit Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quadrature Relaxation Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 High Level Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Quadrature Relaxation Oscillator without Mismatches . . . . . . . . . . . . . . . . . . . . 3.3.3 Quadrature Relaxation Oscillator with Mismatches . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phase-Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Phase-noise in a Single Relaxation Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Phase-noise in Quadrature Relaxation Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 38 38 39 41 41 42 46 54 56 56 60 61

3.1 Introduction This chapter is dedicated to the study of quadrature relaxation oscillators, which consist of two cross-coupled RC relaxation oscillators [9]. In this book we will use interchangeably the designations cross-coupled relaxation oscillator and quadrature relaxation oscillator. Both single relaxation oscillators and the technique of synchronously coupling relaxation oscillators have been known for some time [8–10, 44, 47–51], but their research is still at an initial stage. We present a detailed study of the cross-coupled oscillator using a structured design approach: we first represent the oscillator at a high level, with ideal blocks, and then we study the oscillator at circuit level. This chapter can be divided into two main parts. In the first part we review the basic aspects of single relaxation oscillators and of cross-coupled relaxation oscillators. We present a detailed study of the effect of mismatches on the output voltage and period of oscillation, and we calculate the quadrature error. This analysis is rigorous for low frequency. At high frequency several other effects exist, which are very difficult to include in simple and tractable equations. Thus, at high frequency L.B. Oliveira et al., Analysis and Design of Quadrature Oscillators,  C Springer Science+Business Media B.V. 2008

37

38

3 Quadrature Relaxation Oscillator

we only show how the quadrature relaxation oscillator will react to mismatches (by changing the amplitudes) in order to preserve an accurate quadrature relationship. The second part of this chapter is dedicated to the study of the oscillator phasenoise. We identify the oscillator noise sources and we analyse the phase-noise of single relaxation oscillators. The analysis of cross-coupled oscillators is rather complicated, but a simple qualitative argument indicates that the coupling reduces the phase-noise (as opposed to what happens in coupled LC oscillators). This is demonstrated by simulations.

3.2 Relaxation Oscillator 3.2.1 High Level Model Figure 3.1a shows the block diagram of a relaxation oscillator, which can be modelled using an integrator and a Schmitt trigger. The Schmitt-trigger is a memory element, and controls (switches) the sign of the integration constant. The oscillator waveforms are presented in Fig. 3.1b: the square waveform is the Schmitt-trigger output, and the triangular waveform is the integrator output.

Schmitt-trigger

Integrator

vINT

vST a)

vINT

Amplitude

vST

Time b)

Fig. 3.1 Relaxation oscillator: (a) block diagram; (b) oscillator waveforms

3.2 Relaxation Oscillator

39

3.2.2 Circuit Implementation To implement the oscillator at very high frequencies we need a circuit as simple as possible (Fig. 3.2). Thus, we should substitute the integrator and the Schmitt trigger by simple circuits that ensure some correspondence between the high level and the circuit level. The integrator is implemented simply by a capacitor (Fig. 3.3); its input is the capacitor current (i C ) and the output is the capacitor voltage (νC ). This voltage is the input of the Schmitt-trigger (Fig. 3.4), the output of which is i C . The transfer characteristic of the Schmitt-trigger is shown is Fig. 3.4b. It is assumed that the switching occurs abruptly when the sign of νBE1 − νBE2 changes. Using this approach we have justified the implementation of the known circuit presented in Fig. 3.2 [8] and its relationship with the high level diagram (Fig. 3.1). This circuit implementation is the simplest and can be used for RF applications. Although in terms of the model of Fig. 3.1 the Schmitt-trigger output is i C , it is convenient to use as the oscillator output the voltage νOUT = ν1 − ν2 [8], with an amplitude of 4IR, as shown in Fig. 3.5. However, this is only valid for low oscillation VCC

R

R

v1

v2

Q1

Q2 C

vC I

I

Fig. 3.2 Relaxation oscillator implementation, suitable for high frequency

iC

vC (V )

C

vC

I C

a)

Fig. 3.3 (a) Integrator implementation. (b) Integrator waveforms

t(s)

b)

40

3 Quadrature Relaxation Oscillator VCC

R

R v1

v2

iC = i1 – I

− 2 RI

I Q1

2 RI

Q2

vC i1

iC

−I

i2 +

vC

–

I

I

a)

b)

Fig. 3.4 Schmitt-trigger: (a) circuit implementation; (b) transfer characteristic

Fig. 3.5 Relaxation oscillator waveforms

Amplitude

[V ]

ν OUT

ν INT

2IR

t(s)

–2IR

frequencies; at very high frequencies the outputs are approximately sinusoidal (the harmonics are filtered out by the circuit parasitics) with an amplitude lower than 4IR. A circuit with MOS transistors has the same performance as described above. The circuit in Fig. 3.2 is one possible implementation. A more complex circuit with a more straightforward correspondence between the blocks and their circuit realization was presented in [52]; however, in this case the maximum available frequency is reduced, and the noise, area, and power consumption are higher. This oscillator integration constant is I/C, and the amplitude is 4IR. Thus, the oscillation frequency is: f0 =

I 1 = 2C(4R I ) 8RC

(3.1)

3.3 Quadrature Relaxation Oscillator

41

3.3 Quadrature Relaxation Oscillator 3.3.1 High Level Model In this section we show how to employ two relaxation oscillators to provide quadrature outputs. If we add a soft-limiter (an amplifier with saturation) after the integrator, as represented in Fig. 3.6a, we obtain a new output, with 90◦ phase difference with respect to the Schmitt-trigger output (Fig. 3.6b). By increasing the limiter gain this new output will be closer to a square signal with 90◦ phase shift (with infinite gain, the limiter becomes a hard-limiter, and the output is square). The circuit of Fig. 3.6a, with this 90◦ out of phase output, is itself a quadrature generator, but it has an open-loop structure, in which the output signals have different paths, so there is an error in the quadrature relationship. However, the softlimiter output can be used to synchronize a second relaxation oscillator. Coupling two oscillators using this technique leads to a cross-coupled relaxation oscillator in a feedback structure (Fig. 3.7) [8]. In the cross-coupled oscillator there is not a master and a slave oscillator: both oscillators trigger each other in a balanced structure. The two oscillators lock at

Integrator

Schmitt-trigger

vINT Soft-limiter

vSL a)

vINT

Amplitude

vSL

Fig. 3.6 (a) Relaxation oscillator and a soft-limiter. (b) Soft-limiter input and output

Time b)

42 Fig. 3.7 Cross-coupled relaxation oscillator

3 Quadrature Relaxation Oscillator

Schmitt-trigger

Integrator

vINT1

vST1

v1

1

1 Soft-limiter

vSL1 1 Soft-limiter

2

vSL2 Schmitt-trigger

Integrator

v2 2

vINT2

2

vST2

the same frequency, and the two Schmitt-trigger outputs have inherently 90◦ phase difference (Fig. 3.8a). Figure 3.8b shows the effect of adding the soft-limiter output of one oscillator to the integrator triangular signal of the other. The transition of each oscillator is now defined by a signal with a steeper slope, which means that the switching time is less sensitive to noise. Thus, to minimize the influence of noise on the transition time we should increase the soft-limiter gain. Note that when the two oscillators are equal, the amplitudes and the frequencies of each individual relaxation oscillator are not changed due to coupling.

3.3.2 Quadrature Relaxation Oscillator without Mismatches The cross-coupled relaxation oscillator is implemented with two relaxation oscillators, which are cross-coupled using, as coupling blocks (soft-limiters), differential pairs that sense the capacitor voltage and have the differential output connected to the other oscillator, as shown in Fig. 3.9 [8]. The soft-limiter output is a differential current, which is added at the collector nodes. The effect of this current is to change the input switching levels of the Schmitt-trigger. Thus, this is equivalent to adding a voltage signal at the Schmitt-trigger input, as indicated in the block-diagram of Fig. 3.7. In this section we assume that the coupled oscillators are equal, without mismatches: C1 = C2 = C, I1D = I1C = I2D = I2C = I , and I SL1 = I SL2 = I SL . Each of the coupled oscillators in Fig. 3.9 can be studied as a relaxation oscillator with two extra current sources, which are responsible for the coupling action. For instance, the two current sources i SL1 and i SL2 are provided by the soft-limiter circuit driven by the second oscillator, i.e., one oscillator is synchronously switched (triggered) by the other oscillator.

3.3 Quadrature Relaxation Oscillator

vST1

Amplitude

Fig. 3.8 (a) Integrator output; (b) Schmitt-trigger input with a steeper slope in the transition region

43

Amplitude

Time

vST2

Time a)

v1

Amplitude

vINT1

Time b)

We assume that there are no mismatches between the two oscillators and that the switching occurs instantly when there is a transition of the capacitor voltages by zero. The transistors are assumed to act as switches, which is a good approximation for bipolar transistors; this is also valid for MOS implementations (with high W/L transistors). The waveforms are shown in Fig. 3.10. In this analysis we show why the introduction of coupling will not change the amplitudes of νC1 and νC2 and the oscillation frequency, with respect to the isolated oscillators. In order to determine the amplitudes νC1 and νC2 , we must determine their maximum and minimum values. Due to the oscillator symmetry we will only do the calculations for νC1 (for the amplitude of νC2 the results are the same).

44

3 Quadrature Relaxation Oscillator VCC R

R

R

v1

R

v3

v2

iSL1

iSL 2 Q1

v4 iSL 4

iSL 3

Q3

Q2

Q4 C2

C1

vC1

vC 2

I1C

I1D

QSL

I 2C

I2D

QSL

QSL

2 I SL1

QSL

2 I SL 2

Fig. 3.9 Circuit implementation of a quadrature relaxation oscillator

To calculate the amplitude of νC1 we must consider the two extremes, at instants t1 and t3 in Fig. 3.10. We consider that the oscillator is in steady-state (we do not study the transient regime), and we assume that νC1 and νC2 are in quadrature; the waveform in advance of 90◦ can be either νC1 or νC2 , depending on which of the two coupling connections is direct and which is reversed. Note that the oscillators can lock in phase or in quadrature, depending on the sign of the summations at the input of the Schmitt-trigger (Fig. 3.7), i.e. the polarity of connection of the soft-limiters in Fig. 3.9. The polarity shown in Fig. 3.9 produces quadrature oscillations with νC2 in advance. We will start the analysis by considering that Q 2 is on and Q 1 is off. When νC2 (t) goes through zero (instant t1 in Fig. 3.10) i SL1 decreases and i SL2 increases. We have 

ν1 = VCC − i SL1 R ν2 = VCC − 2I R − i SL2 R

(3.2)

The transistors change state when ν B E1 = VBE O N , and immediately before the switching occurs, νC1 = ν2 − ν1 + ν B E2O N − ν B E1O N

3.3 Quadrature Relaxation Oscillator Fig. 3.10 Waveforms in a symmetric quadrature oscillator (without mismatches)

45 t0

t2

t1

T3

T2

T1

t3

t4 T4

vC 1 slope

2 RI

I C

− 2 RI

Q1off Q3 on

Q1on Q3 on

Q1on Q3 off

Q1off Q3 off

vC 2

2 RI

slope − 2 RI

I C

v1 − v2

− 2 RI

2 R( I + I SL )

− 2 R( I + ISL )

v3 − v4 2 R( I + I SL )

− 2 R( I + I SL )

Assuming that ν B E2O N ≈ ν B E1O N , νC1 ≈ ν2 −ν1 = VCC −2I R−i SL2 R−[VCC −i SL1 R] = −2I R−R(i SL2 −i SL1 ) (3.3)

Since it is assumed that the switching is provided by a vanishingly small value of i SL2 − i SL1 the minimum value of νC1 is −2I R. Considering now the second transition of νC2 by zero (instant t3 in Fig. 3.10) i SL1 increases and i SL2 decreases. With Q 1 on and Q 2 off we have:

46

3 Quadrature Relaxation Oscillator



ν1 = VCC − 2I R − i SL1 R ν2 = VCC − i SL2 R

(3.4)

This state will be over (transistors will change state), when νC1 = ν2 − ν1 + ν B E2O N − ν B E1O N νC1 = VCC − i SL2 R − [VCC − 2I R − i SL1 R] = 2I R − R(i SL2 − i SL1 )

(3.5)

Since the switching occurs with a small value of R(i SL2 − i SL1 ), the maximum value of νC1 is νC1 = 2I R. Using (3.3) and (3.5), we can conclude that the amplitude of νC1 does not change due to the coupling. The same result can be determined for νC2 , by doing the calculations at the instants t2 and t4 of Fig. 3.10. With mismatches between the two oscillators (e.g., C1 = C2 ) one oscillator provides a trigger signal to the other, due to the coupling, and tries to modify the amplitude and period of the other oscillator. At the end of a transient period both relaxation oscillators will have different amplitudes but the same frequency, different from f 0 . This change of amplitude and frequency due to mismatches will be analysed in detail in the next section.

3.3.3 Quadrature Relaxation Oscillator with Mismatches In this section we derive the amplitudes of νC1 and νC2 and the oscillation period for the circuit in Fig. 3.9, with mismatches, and we calculate the quadrature error. This analysis is important to understand how the amplitudes change in order to preserve the quadrature relationship, which explains why this oscillator has very accurate quadrature. In the following derivation we assume that the collector resistors R are identical in the two coupled oscillators. We consider that the trigger in one oscillator occurs instantly when the capacitor voltage νC in the other oscillator goes through zero. In reality, the switching occurs after a small delay, which we assume to be much lower than the period. This approximation is valid for small relative mismatches in the circuit components (⌬C/C << 1 and ⌬I /I << 1) and for higher mismatches if we increase the coupling gain. To simplify the calculations we assume that the oscillators are initially identical, without mismatches, and the capacitor voltages (νC1 and νC2 ) are triangular waveforms shifted by 90◦ . The waveform in advance can be either νC1 or νC2 , depending on the oscillator that has the higher stand-alone oscillation frequency. In the following analysis, we consider the general case in which C1 and C2 are different, and the currents I1C , I2C , I1D , and I2D are all different (Fig. 3.11). We consider that the new values of capacitances, and charge and discharge currents, occur by deviation from their ideal values. When all changes have sequentially been considered we redo the calculations and obtain the same amplitude at the beginning and at the end of the period, which shows that the circuit is in steady state oscillation. The mismatches in capacitances and in the currents are:

3.3 Quadrature Relaxation Oscillator Fig. 3.11 Waveforms in a quadrature oscillator with mismatches

47 t0

T1

t3

t2

t1

t4

T3

T2

T4 vC1max

vC1 slope

I1C C1

slope

I1D C1

vC1min

vC2max

vC2

slope

I2C C2

slope

I2D C2

vC2min

− 2 RI

v1 − v2

2R(I + ISL2)

–2R(I + ISL2) v3 − v4

2R(I + ISL1)

–2R(I + ISL1)

 C1 = C + ⌬C C2 = C − ⌬C

(3.6)

⎧ ⎪ ⎪ I1C = I + ⌬I1 ⎪ ⎨ I = I − ⌬I 1D 1 ⎪ I2C = I + ⌬I2 ⎪ ⎪ ⎩ I2D = I − ⌬I2

(3.7)

We consider that at, t = 0, νC2 (0) = − 2I R where I is the same as in equation (3.7). The charging current at this instant becomes I2C . For t > 0: νC2 (t) = −2I R +

I2C t C2

(3.8)

48

3 Quadrature Relaxation Oscillator

and from νC2 (t1 ) = 0 we obtain: t1 = 2RC2

I

(3.9)

I2C

The capacitor in the first oscillator, with capacitance C, continues to discharge after t = 0, with current I. At the instant t1 the discharge finishes (the required change of the voltage at the collectors of the first relaxation oscillator will be provided instantly by the currents i SL1 and i SL2 ; this approximation is valid for a high coupling gain. I C2 I νC1 (t1 ) = − t1 = −2I R C C I2C

(3.10)

After t = t1 the capacitance of the first relaxation oscillator becomes C1 . Simultaneously the charging current becomes I1C . For t > t1 : νC1 (t) = −2I R

C2 I I1C + (t − t1 ) C I2c C1

(3.11)

Since νC1 (t2 ) = 0, t2 = 2RC2

I I2C

 1+

C1 I C I1C

 (3.12)

and replacing (3.12) in (3.8), νC2 (t2 ) = 2I R

C1 I C I1C

(3.13)

At t = t2 the discharge current of the second relaxation oscillator becomes I2D , νC2 (t) = 2I R

C1 I I2D − (t − t2 ) C I1C C2

(3.14)

and, since νC2 (t3 ) = 0  t3 = 2RC2

I I2C

C1 + C



I2 I2 + I1C I2C I1C I2D

 (3.15)

From (3.11): νC1 (t3 ) = −2I R

C2 I I1C + (t3 − t1 ) C I2C C1

(3.16)

3.3 Quadrature Relaxation Oscillator

49

and from (3.15) and (3.9): t3 − t1 = 2RC2

C1 I C I1C



I I2C

+

I

 (3.17)

I2D

Replacing (3.17) in (3.16), leads to: C2 I C I2D

νC1 (t3 ) = 2R I

(3.18)

We assume that when t = t3 the discharge current in the first relaxation oscillator becomes I1D . For t > t3 : νC1 (t) = 2R I

C2 I I1D − (t − t3 ) C I2D C1

(3.19)

Since νC1 (t4 ) = 0,  t4 = 2RC2

I

+

I2C

C1 C



I2 I2 I2 + + I1C I2C I1C I2D I1D I2D

 (3.20)

From (3.14): νC2 (t4 ) = 2I R

C1 I I2D − (t4 − t2 ) C I1C C2

(3.21)

and from (3.20) and (3.12) C1 I t4 − t2 = 2RC2 C I2D



I I1D

+

I I1C

 (3.22)

Thus, νC2 (t4 ) = −2I R

C1 I C I1D

(3.23)

Now all parameters are introduced, and to find the steady-state values we must use νC2 (t4 ) as the new initial value and repeat the calculations using C1 , C2 , and I1C , I1D , I2C , I2D as the parameters of the relaxation oscillators. Following the previous sequence of calculations we obtain:  t1 = 2R

C1 C2 C



I2 I1D I2C

(3.24)

50

3 Quadrature Relaxation Oscillator

 t2 = 2R

C1 C2 C





t3 = 2R

C1 C2 C

 t4 = 2R

C1 C2 C



I2 I2 + I1C I2C I1D I2C

 (3.25)

I2 I2 I2 + + I1C I2C I1D I2C I1C I2D



I I1C

+

I



I1D

I I2C

+

I

 (3.26)

 (3.27)

I2D

The oscillator is in steady state oscillation (the starting voltage and the final voltage in one period is the same), and the final waveforms are represented in Fig. 3.12. After determining the steady-state waveforms, and their zero crossings, we can obtain the time intervals that are used to calculate the duty-cycle, quadrature relationship, and oscillator frequency. T1 = t1

(3.28)

T2 = t2 − t1

(3.29)

T3 = t3 − t2

(3.30)

T4 = t4 − t3

(3.31)

T = T1 + T2 + T3 + T4

(3.32)

vC1,2

C I 2IR 2 C I2D

vC1

C I 2IR 1 C I1C

vC2 t1

Fig. 3.12 Oscillator steady-state waveforms

C I −2IR 1 C I1D

t2 C I −2IR 2 C I2C

t3

C I −2IR 1 C I1D

t4

t

3.3 Quadrature Relaxation Oscillator

51

The duty-cycles are defined as: dc1 =

T1 + T2 I1C I2D + I1D I2D = T (I1C + I1D )(I2C + I2D )

(3.33)

dc2 =

T1 + T4 I1C I2D + I1C I2C = T (I1C + I1D )(I2C + I2D )

(3.34)

From the above equations we can conclude that the duty-cycle is 50% only if the currents that charge and discharge the capacitors are equal. The duty-cycle does not depend on the capacitance values. The phase difference (␾) in a square-wave without 50% duty-cycle is not clearly defined. We have used the definition shown in Fig. 3.13. A phase difference applies to sinusoidal waveforms; however, the phase difference of two square waveforms, can be defined as shown in Fig. 3.13. 

T2 + T3 T1 + T2 − ␾ = T1 + 2 2 =␲



2␲ T

I1C I2D + I1D I2C T1 + T3 =␲ T (I1C + I1D )(I2C + I2D )

(3.35)

This equation takes into account the mismatches of all the current sources. Using (3.7) and replacing in (3.35), ␾=␲

(I + ⌬I1 )(I − ⌬I2 ) + (I − ⌬I1 )(I + ⌬I2 ) (I + ⌬I1 + I − ⌬I1 )(I + ⌬I2 + I − ⌬I2 ) (3.36)

  ␲ I 2 − ⌬I1 ⌬I2 ␲ ⌬I1 ⌬I2 = = 1− 2 I2 2 I2 T1

T2

T3

T4

φ 0

Fig. 3.13 Definition of phase difference of two square-waves

T1 + T2

2

T1 +

T2 + T3

2

52

3 Quadrature Relaxation Oscillator

Equation (3.36) is valid for small relative mismatches; it is still for higher mismatches if the coupling gain is high. Equation (3.36) proves that the mismatches in the currents have a second order effect on the quadrature relationship. With high relative mismatches and low coupling gain other terms must be added to (3.36). For outputs exactly in quadrature, the oscillators may have different capacitances but, from (3.35), the following conditions should be satisfied I1C = I1D = I1

(3.37)

I2C = I2D = I2

(3.38)

and

Only in this case we obtain perfect quadrature ␾=

␲ 2

(3.39)

Finally, we can determine the equation for the oscillation frequency. f =

C I1C I1D 1 I2C I2D 1 = T 2R I 2 C1 C2 (I1C + I1D ) (I2C + I2D )

(3.40)

Without mismatches f = f 0 , with f 0 given by (3.1). From the previous derivations we can obtain the maximum and minimum values of the voltages νC1 and νC2 : C2 I C I2C

(3.41)

C2 I C I2D

(3.42)

νC1min = νC1 (t1 ) = −2R I

νC1max = νC1 (t3 ) = 2R I

νC2min = νC2 (t4 ) = −2R I

νC2max = νC2 (t2 ) = 2R I

C1 I C I1D

C1 I C I1C

(3.43)

(3.44)

It is interesting to analyze how the relaxation oscillators adjust the amplitude of the capacitor voltages to preserve the quadrature. For example, if C2 decreases, we can see from (3.41) to (3.44) that the amplitude of νC1 decreases. As a consequence the first relaxation oscillator becomes faster (the oscillation frequency increases) and is able to follow the second relaxation oscillator. A similar analysis can be done for the variation of the charge and discharge currents. If these currents increase in the second relaxation oscillator, the first relaxation oscillator responds by reducing

3.3 Quadrature Relaxation Oscillator

53

the oscillation amplitude to follow the second relaxation oscillator. Thus, the oscillator changes the amplitude and the oscillation frequency in order to preserve the quadrature relationship. Finally, the following interesting observation can be done. We consider that C1 = C2 = C and that in the first relaxation oscillator I1C = I1D = I . If I2C = I +⌬I and I2D = I − ⌬I in the second relaxation oscillator, the amplitude of νC2 is preserved and νC2max = |νC2min | = 2I R, while from (3.41) and (3.42): |νC1min | = 2I R

I I + ⌬I

(3.45)

νC1max = 2I R

I I − ⌬I

(3.46)

The voltage νC1 has now a DC component with value: VC1 =

νC1max − |νC1 min | 2

(3.47)

⌬I ≈ 2R⌬I = 2I R 2 I − (⌬I )2 2

In this case the oscillator preserves the quadrature relationship by changing the amplitude and the frequency, and by adding a DC component to νC1 .

1

vINT1

–1

1

–1

1

–1

1

1

2 vINT2

Fig. 3.14 Cross-coupled oscillator block diagram with variable output levels in one Schmitt-trigger

2

–

G –G

54

3 Quadrature Relaxation Oscillator

3.3.4 Simulation Results We simulated the quadrature cross-coupled oscillator at a high level using ideal blocks with MATLAB, to confirm the amplitudes change predicted by the theoretical analysis. As shown in the block diagram of Fig. 3.14, we change the integration slope of the second oscillator by changing the Schmitt trigger outputs levels. When we increase the integration slopes in the second oscillator, the amplitude of the first oscillator decreases to follow the frequency of the second oscillator (Fig. 3.15a), and if we have different positive and negative slopes, a DC offset appears (Fig. 3.15b). These simulations do not validate the theoretical analysis of the

1.5

ΔV

1

vINT2

vINT1 Amplitude (V)

0.5

0

−0.5

−1 −1.5 100

ΔV 101

102

103

104

105 106 Time (s) (a)

107

108

109

110

107

108

109

110

1.5

ΔV 1

vINT2 vINT1

Amplitude (V)

0.5

0

−0.5

ΔV

−1 −1.5 100

101

102

103

104

105 106 Time (s) (b)

Fig. 3.15 Effect of changing the integrator slopes in oscillator 2: (a) Increasing both slopes by 10%; (b) Increasing the positive slope by 10% and decreasing the negative slope by 10%

3.3 Quadrature Relaxation Oscillator

55

previous section, but they confirm that the amplitudes change due to mismatches, as predicted by the theoretical analysis. In order to confirm the theoretical analysis at circuit level, we designed a 2.4 GHz oscillator using a 0.35 ␮m CMOS technology (the circuit of Fig. 3.9, but with MOS transistors). To achieve that very high frequency, the circuit was designed with the following parameters: R = 100 ⍀, C1 = C2 = 420 f F, I1 = I2 = 3 mA, I SL = 1 mA and VCC = 3 V. The transistor dimensions are: 200 ␮m/0.35 ␮m for the M transistors and 80 ␮m/0.35 ␮m for the M SL transistors. The theoretical analysis presented is rigorous for low frequency, with triangular waveforms; at high-frequency several other effects are present, which are very difficult to quantify (the first RF transistor models became available only recently, and they have some limitations; research is still active in this area). In this RF circuit implementation many parasitics are present; the output is approximately a sinewave and not a triangular waveform, due to the filtering action performed by the parasitics. The simulation results show the amplitude changes (Fig. 3.16) and the DC offset (Fig. 3.17), as expected from the theoretical analysis. The theoretical amplitudes are 2I R = 600 mV and the simulated amplitudes are about 400 mV. Although the absolute value of the amplitude has a significant difference, due to high frequency effects, the relative changes show a good agreement with the theory: (1) If we change the capacitors by 10% (small relative mismatches in capacitors, ⌬C/C = 0.1 << 1), the amplitudes should also change 10%. The simulated

vINT2 vINT1

ΔV

ΔV

Fig. 3.16 Effect of decreasing C2 by 10%

56

3 Quadrature Relaxation Oscillator

vINT1

ΔV

vINT2

ΔV

Fig. 3.17 Effect of changing the current values by 10% (I2C = 3.3 mA and I2D = 2.7 mA)

values are VINT1 = 0.352 mV and VINT2 = 0.381 mV, which correspond to a relative change of about 8%. (2) By changing the currents in the second oscillator by 10% (I2C = 3.3 mA and I2D = 2.7 mA), with small relative mismatches ⌬I /I = 0.1 << 1, we expect a DC offset of about 10%. The simulated DC offset is about 49 mV, which corresponds to a relative change in the amplitude of about 12%.

3.4 Phase-Noise This section is dedicated to the study of the relaxation oscillator phase-noise and the influence on phase-noise of cross-coupling two relaxation oscillators. We review the basic aspects of phase-noise in a single relaxation oscillator [1, 36, 44, 48, 50, 53–55]. We argue that the coupling should reduce the phase-noise of cross-coupled relaxation oscillators, and we demonstrate by simulations that this is so.

3.4.1 Phase-noise in a Single Relaxation Oscillator The noise in electronic circuits is usually described by the noise power spectral density S(xn ), where xn is a noise variable (usually either voltage or current). In the noise analysis of linear systems the following result is used [56, 57]. Assuming that an input x(t) produces an output y(t), and that the corresponding transfer

3.4 Phase-Noise

57

function is H (s) = Y (s)/ X (s), a noise source xn with spectral density S(xn ) at the input produces an output noise yn with spectral density S(yn ) = |H (s)s= j␻ |2 S(xn )

(3.48)

Usually there is more than one noise source in a network. In this case the following result concerning random variables is used. If a noise variable y(t) is a linear combination of different random variables x1 (t) and x2 (t), y(t) = a1 x1 (t) + a2 x2 (t)

(3.49)

and if x1 and x2 are independent, the relationship between the spectral densities is: S(y) = a1 2 S(x1 ) + a2 2 S(x2 )

(3.50)

Note that if the variables are correlated (this is not considered in this book), there is an additional term in the above equation, and the derivations become rather complicated. In the time domain the oscillator output (νOUT ) is: νOUT = Vom cos[␻0 t + ␪(t)]

(3.51)

where Vom is the oscillator amplitude and ␪(t) contains the oscillator phase-noise. A carrier of amplitude Vom , modulated in phase by a sinusoidal signal of frequency f m can be represented by [58]:   ⌬ f pk sin(␻m t) νOUT = Vom cos (␻0 t) + fm

(3.52)

where ⌬ f pk is the peak frequency deviation. In this book we are interested in the study of oscillators in which the peak frequency deviation is the deviation of the oscillation frequency due to low frequency noise sources (⌬ f 0 = ⌬ f pk ). As shown in textbooks on FM theory [58–60], for f m << f 0 the phase modulation results in frequency components on each side of the carrier, with amplitude Vom ⌬ f 0 /2 f m :   ⌬ f0 [cos(␻0 + ␻m )t − cos(␻0 − ␻m )t] νOUT ≈ Vom cos(␻0 t) + 2 fm ⌬ f0 νOUT | f = f0 + fm = Vom cos(␻0 + ␻m ) (3.53) 2 fm Using equation (3.48) and the FM theory for low frequency noise sources with f m << f 0 we obtain S(νOUT )| f = f0 + fm

   Vom 2   S(⌬ f 0 ) = 2 fm 

(3.54)

58

3 Quadrature Relaxation Oscillator

Fig. 3.18 Noise analysis for an oscillator

xn (t)

H(s)

Δf0 (t)

The spectral density of the phase-noise, L( f m ), is defined as the ratio of the noise power in a 1 Hz bandwidth at a distance f m from the carrier relative to the carrier power. L( f m ) =

S(νOUT )| f = f0 + fm 1 V 2 2 om

=

1 S(⌬ f 0 ) 2 f m2

(3.55)

Considering (Fig. 3.18) that a noise source xn (t) with a spectral density S(xn ) in an oscillator will originate a frequency shift with spectral density S(⌬ f 0 ), it follows from equation (3.48): S(⌬ f 0 ) = |H ( j␻)|2 S(xn )

(3.56)

where H(s) relates the transforms of variables ⌬ f 0 and xn . By taking into account (3.55) and (3.56), L( f m ) =

|H ( j␻)|2 S(xn ) 2 f m2

(3.57)

We consider the relaxation oscillator at a high level and assume that the high level model is that in Fig. 3.19: the integrator is a capacitor, which is charged and discharged by a current source controlled by the Schmitt-trigger. The oscillation frequency of the relaxation oscillator in Fig. 3.19 is: f0 =

I 2C V

(3.58)

where I is the current that charges and discharges the capacitance C, and V is the Schmitt-trigger difference of threshold voltages. The noise sources in Fig. 3.19 are the equivalent noise current i n in parallel with current source I, and the equivalent noise voltage νn in series with the Schmitt-trigger input. The noise sources modulate the oscillation frequency, thus creating phase-noise. The frequency shift due to i n , assuming that i n is approximately constant during one period, i.e., assuming low-frequency noise, is obtained from (3.58): ⌬ f0 = and using (3.57)

in in = f0 2C V I

(3.59)

3.4 Phase-Noise

59

Fig. 3.19 Relaxation oscillator with noise sources

VCC

in

I Schmitt-trigger

vn

C

V

L( f m ) =

S(i n ) 2I 2



2

f0 fm

(3.60)

The frequency shift due to νn , using (3.58), is ⌬ f0 = −

I f0 νn = − νn 2C V 2 V

and using (3.57) S(νn ) L( f m ) = 2V 2



f0 fm

(3.61)

2 (3.62)

Equations (3.60) and (3.62) apply to low-frequency noise. However, we must also take into account the high frequency noise: the switching, which is inherent to relaxation oscillators, produces a mixing effect that will “fold back” high frequency noise components, resulting in low-frequency phase-noise [35, 48, 51, 55]. For the case of high frequency current noise components, the resulting phasenoise is filtered out due to the integrating capacitor. Thus, the resulting phase-noise is still given by (3.60). The high frequency voltage noise components will be dominant and produce the main contribution to the oscillator phase-noise [35, 48, 51]. Assuming that νn is white noise, it can be demonstrated [48,51] that the resulting phase-noise is L( f m ) = 2␣

4S(νn ) 2V 2



f0 fm

2 (3.63)

where ␣=

Bc 2 f0

(3.64)

60

3 Quadrature Relaxation Oscillator

and where Bc is the bandwidth for which there is significant noise conversion (this depends on the circuit implementation).

3.4.2 Phase-noise in Quadrature Relaxation Oscillators As indicated above by Fig. 3.8b a phase-noise reduction with strong coupling is expected, since the switching of each oscillator is defined by a signal with a steeper slope (by increasing the coupling current we increase the slope); this ensures that the circuit is less sensitive to noise in the transition points that define the oscillator frequency. We perform circuit level simulations of a 5 GHz oscillator (with and without coupling) to demonstrate that the coupling improves the phase-noise performance. It can be shown that the phase-noise of N coupled oscillators is reduced by a factor 1/N with respect to a single oscillator [54,61]. For two coupled oscillators (RC and LC), this improvement of 3 dB was confirmed in [34, 44, 62–64]. This result is valid for two coupled oscillators with low coupling gain; if we increase the coupling gain, a further reduction of the oscillator phase-noise is expected in relaxation RC oscillators (in LC oscillators there is a noise increase as will be shown in Chapter 5). At circuit level the increase of coupling gain is performed by increasing the softlimiter bias current. We use a quadrature oscillator designed in a 0.18 ␮m CMOS technology, for an oscillation frequency of 5 GHz. The circuit parameters are R = 100 ⍀, (W/L) = 100 ␮m/0.18 ␮m for M transistors, (W/L) = 100 ␮m/0.18 ␮m for M SL transistors, –50 1) Stand alone 2) ISL = 0.1 mA 3) ISL = 3 mA

–60

Phase Noise (dBc/Hz)

–70 –80 –90 –100 –110 –120 –130 –140 –150 4 10

5

10

6

10 Offset Frequency (Hz)

7

10

Fig. 3.20 Influence of coupling on the phase-noise of a relaxation oscillator

8

10

3.5 Conclusions

61

C = 300 fF, I = 3 mA, and I SL = 0.1 mA (low coupling gain) or I SL = 3 mA (high coupling gain). The simulations are done with spectreRF considering ideal current sources. Figure 3.20 shows the simulated phase-noise for a stand-alone relaxation oscillator, and for coupled relaxation oscillators. We can observe that the phase-noise of −113.85 dBc/Hz @ 10 MHz is improved by 3 dB with weak coupling. Increasing I SL from 0.1 mA to 3 mA gives a further reduction of the phase-noise of about 6.2 dB, bringing the total reduction to 9.2 dB. In Chapter 7 we will present experimental results for this oscillator, and we will compare it with an LC cross-coupled oscillator designed for the same frequency and using the same technology.

3.5 Conclusions In this chapter we presented a detailed study of the quadrature relaxation oscillator concerning its key aspects: oscillation frequency, output amplitude, and quadrature relationship. The phase-noise is investigated by simulation. An oscillator consisting of two relaxation oscillators that are cross-coupled using two coupling blocks has outputs in very accurate quadrature. The variation of parameters in one oscillator is compensated by a variation of the amplitude of the other. Moreover, we show that a difference between the charge and discharge currents in one relaxation oscillator results in a DC component in its capacitor voltage. Thus, the quadrature is preserved, and the effect of mismatches is automatically compensated. We identify the noise sources and derive an equation for the phase-noise of a single relaxation oscillator at a high level. To investigate the phase-noise of quadrature cross-coupled relaxation oscillators we simulate a 5 GHz circuit, which confirms that there is a reduction of the oscillator phase-noise due to coupling. From our study we can conclude that the coupling block has a strong influence on the quadrature oscillator performance. Increasing the gain of the coupling block will improve the oscillator performance: the effect of mismatches on the quadrature error is reduced, and becomes a second order effect, as shown by (3.36); the oscillator phase-noise is also reduced. Thus, to improve the performance of a quadrature relaxation oscillator, the value of the coupling gain should be increased, but this has a cost in terms of power consumption.

Chapter 4

Quadrature Oscillator-Mixer

Contents 4.1 4.2

4.3 4.4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . High Level Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Ideal Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Effect of Mismatches and Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Circuit Level Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 64 64 67 75 79

4.1 Introduction In this chapter we study the possibility of including the mixing function in the oscillator, thus avoiding the use of external mixers. Conventional mixers have an open-loop structure, in which the output is obtained by multiplication of an LO signal with an input signal (RF signal, in the receiver path). For quadrature modulation/demodulation two independent mixers are necessary, and accurate matching of circuit elements is required to avoid quadrature errors. In the previous chapter, we proved that the coupling reduces the effect of mismatches, providing very accurate quadrature of the outputs. In this chapter we show that accurate quadrature is maintained when the mixing function is integrated in the oscillator, without extra cost, either in area or in power consumption. In [15, 65] a theoretical study of the quadrature relaxation oscillator was presented using Matlab and with low frequency validation, and in [66] the oscillatormixer concept was presented and demonstrated using Matlab. In this chapter we demonstrate the same concept with a high frequency circuit implementation, and we study the influence of the mixing function on the quadrature oscillator performance. This can be seen as an extension of the results presented in [15, 65]. In this chapter we present a high level theoretical study of the influence of mismatches and delays on the duty-cycle, oscillation frequency, and quadrature relationship in the oscillator-mixer. This study is done considering the coupling block as a linear amplifier, because at the switching points the coupling block is in its linear region. For validation at circuit level we designed a 2.4 GHz quadrature L.B. Oliveira et al., Analysis and Design of Quadrature Oscillators,  C Springer Science+Business Media B.V. 2008

63

64

4 Quadrature Oscillator-Mixer

relaxation oscillator-mixer, for which we show that, with mismatches, the oscillatormixer has very accurate quadrature outputs. Therefore, we show that the inclusion of the mixing function in the cross-coupled oscillator does not affect significantly the oscillator good performance.

4.2 High Level Study 4.2.1 Ideal Performance The quadrature relaxation oscillator has accurate quadrature outputs due to feedback, and the components mismatch has only a second order effect on the frequency and phase accuracy [15]. The goal is to obtain the mixing function in this circuit without disturbing the good oscillator performance. We can inject the modulating signal (multiplying the input signal with the oscillator signal) in three different blocks in the model of Fig. 4.1 [66]: 1 – Integrator – If we multiply the input signal by the signal either before or after the integrator, the zero crossings of the oscillator output will change, which will affect its phase and frequency stability. 2 – Schmitt-trigger – If we multiply the input signal with the signal before the Schmitt-trigger or inside of the Schmitt-trigger block (Fig. 3.2), the Schmitttrigger decision levels will be affected, and, therefore, the integrator signals will also be affected. As in the previous case, this will change the frequency and phase of the oscillator. 3 – Soft-limiter – Multiplying the input signal with the signal either before or after the soft-limiter is the only possibility to avoid deteriorating the oscillator performance, because this does not change the timing of the zero-crossings of the soft-limiters outputs, which are responsible for the quadrature relationship and frequency of the oscillator (Fig. 4.1). The soft-limiter output provides a sort of injection locking, by triggering the other oscillator so that it changes at the right Integrator

Schmitt-trigger +

A

Soft-limiter

+

1 –1

A

vf1

–1

1

vOUT

+ +

vf 2 B

Fig. 4.1 Quadrature relaxation oscillator-mixer in an upconversion configuration

B

–

1

+

–1

–1

1

4.2 High Level Study

65

time. Changing the gain of the soft-limiter, or changing its saturation levels, will change only the strength of this triggering, and the performance of the oscillator will not be affected as a first order effect. Thus, this circuit can be used as modulator and demodulator for quadrature signals. In the high-level model of Fig. 4.1 we have basically two possibilities: to add a new input to the soft-limiter block, to control its gain, or to change the soft-limiter saturation levels. The injected signal must be associated to a DC component, to avoid a soft-limiter output change of sign. In the block diagram in Fig. 4.1, two voltage inputs, are added, one with frequency f 1 , in soft-limiter A (I path) and the other with frequency f 2 in the softlimiter B (Q path). In the Matlab block diagram we basically add an ideal multiplier (linear modulation) after the soft-limiter in order to perform the mixing function. We have 

ν f 1 = A + a cos(␻1 t) ν f 2 = B + b sin(␻2 t)

(4.1)

where |a| < A and |b| < B. The soft-limiter is implemented by a differential pair, in which a change of gain is obtained by changing the bias current: we do not need to add any DC value (the DC component is the soft-limiter bias current, which physically cannot be negative). The oscillator output is an approximately square waveform with fundamental frequency ␻0 and with several harmonics at n␻0 (n integer). In the following derivation we will consider only the fundamental frequency ␻0 (the higher harmonics will be filtered out after the mixing). The output of the oscillator-mixer is νOUT = An cos(␻0 t) cos(␻1 t) + An sin(␻0 t) sin(␻2 t) =

1 An [cos(␻0 + ␻1 )t + cos(␻0 − ␻1 )t] 2 1 + An [sin(␻0 + ␻2 )t + sin(␻0 − ␻2 )t] 2

(4.2)

The oscillator-mixer performing upconversion is simulated in Matlab, considering two different modulating signals, normalized to the carrier before mixing. The output is in accordance with (4.2) and is represented in the frequency domain in Fig. 4.2. In a complete transmitter this signal is filtered (removing undesired higher frequencies), amplified by the power amplifier (PA) and transmitted by the antenna. To recover the information we need to perform a downconversion, by using the oscillator-mixer as shown in Fig. 4.3, where the RF input signal is received by the antenna and amplified by a low-noise amplifier (LNA). The RF signal (νIN ) is applied directly to the soft-limiters as in the upconversion case.

66

4 Quadrature Oscillator-Mixer 10

f0

0 –10

Amplitude (dB)

–20 –30

f0 − f1

–40

f0 − f2

–50

f0 + f1 f0 + f2

–60 –70 –80 –90 –100 0.6

0.7

0.8 0.9 1 1.1 1.2 Frequency (Normalized to Carrier Frequency)

1.3

Fig. 4.2 IQ modulation with I at f 1 and Q at f 2 (Matlab simulation)

Considering that νIN is given by (4.2), the outputs of the I and Q channels are, respectively, νI =

1 An cos(␻1 ) + terms with higher frequencies 4

(4.3)

νQ =

1 An sin(␻2 ) + terms with higher frequencies 4

(4.4)

vI

vIN vQ

Fig. 4.3 IQ oscillator-mixer in a downconversion configuration

4.2 High Level Study

67

0 I Signal Q Signal

–10 –20

Amplitude (dB)

–30

f1

f2

–40 –50 –60 –70 –80 –90 –100 –110

0

0.05

0.1

0.15 0.2 0.25 0.3 0.35 Frequency (Normalized to the Carrier)

0.4

0.45

0.5

Fig. 4.4 IQ demodulation with I at f 1 and Q at f 2

To recover the I ( f 1 ) and Q ( f 2 ) signals we only need to apply low-pass filtering to the soft-limiter outputs to remove the high frequency terms in (4.3) and (4.4). To perform correctly the demodulation, avoiding cross-talk between the channels, the demodulator’s oscillation frequency and phase must be the same as those in the modulator. To synchronize the two oscillators, in the modulator and demodulator, a PLL should be used. Traditional PLLs (using a mixer and a low-pass filter) only synchronize the frequency and not the phase, while a charge-pump PLL can synchronize both frequency and phase [1]. Simulation of the complete system with modulation and demodulation, using a PLL to synchronize the oscillators, produces only the f 1 signal in the I path and only the f 2 signal in the Q path, as expected; this is shown in Fig. 4.4.

4.2.2 Effect of Mismatches and Delay In the previous section we have considered that the two coupled oscillators are identical. In this section we will study the oscillator-mixer quadrature relationship and oscillation frequency with mismatches (the amplitude will be considered only in the study at circuit level). In a practical implementation, mismatches between components and other disturbances, such as delays, produce variation of the frequency and phase difference. To investigate the effect of including the mixing in the oscillator, we consider two signals with different frequencies at the input of the soft-limiters, x1 (t) and

68

4 Quadrature Oscillator-Mixer

Fig. 4.5 Quadrature modulation with I signal at frequency f 1 and Q signal at frequency f 2

X aA

X iA

DlA DhA

τA

X oA

X lB

x 1 ( t) x 2 ( t)

τB

X lA –

X iB B

+

X oB

X aB –1

DlB DhB

x2 (t). We assume that the soft-limiters are in their linear region, so any change of the saturation levels is not relevant: what matters is the gain in the linear region. In the following analysis we consider that the soft-limiter acts as a linear amplifier and that the modulating signal changes its gain. The block diagram, with the variables marked, is represented in Fig. 4.5, and the variables are defined in Table 4.1. In the following derivation we consider that the two input frequencies are f 1 << f 0 and f 2 << f 0 , i. e., we assume that the two input signals are approximately constant in one period, which is only valid for the upconversion case (Fig. 4.1). In the case of downconversion (Fig. 4.3) the analysis is much more complicated, because the input frequency is of the same order of the oscillation frequency (this is not studied in the book). In Fig. 4.6 we represent the integrator and Schmitt-trigger waveforms, in quadrature (note that the waveforms are symmetric, which is only valid for equal oscillators and without delays), and we mark the time instants and time intervals used in the analysis. Exact equations for the time intervals T1 , T2 , T3 , and T4 as a function of

Table 4.1 Circuit parameters X i{A,B} X s{A,B} X l{A,B} X a{A,B} {+,−} K i{A,B} G l{A,B} Dl{A,B} Dh{A,B} τ{A,B} x1 , x2

Output of integrator A or B Output of Schmitt-trigger A or B Output of soft-limiter A or B Output of adder A or B Constant of integrator A or B (positive or negative slopes) Gain of soft-limiter A or B Low decision level of Schmitt-trigger A or B High decision level of Schmitt-trigger A or B Delay in cross-coupling path A or B Input signals

4.2 High Level Study

t0

69

t1 T2

T1

t3

t2

T3

t4 T4

T1

Xi{A,B} X iA

t

X iB

Xo{A,B} 1

X oA

t

−1

X oB 1

t

−1 Fig. 4.6 Oscillator waveforms

the circuit parameters are determined using an approach similar to that in [15, 65], where the oscillator was studied without mixing. In order to calculate the four time intervals shown in Fig. 4.6 we need to determinate a system of equations for each of the four states. The equations are obtained using a symbolic algebraic program (Derive ). Determination of T1 We assume that in the state that lasts T1 in Fig. 4.6 the Schmitt-trigger outputs are X o A = 1 and X oB = 1, and therefore, the integrator inputs are positive. The integrator outputs, at t = t1 , considering the effect of delay in the cross-coupling paths of the oscillator system are: X i A (t1 ) = X i A (t0 ) + K i+A T1

(4.5)

X i B (t1 − τb ) = X i B (t0 ) + K i+B (T1 − τb )

(4.6)

70

4 Quadrature Oscillator-Mixer

The soft-limiter B output, responsible for the triggering of oscillator A is modulated by input signal x2 . In the following analysis we consider that the soft-limiter gains are not constant: they are modulated by two input signals x1 (t) and x2 (t) with a frequency much lower than the oscillation frequency. The soft-limiter output value at t1 is: X l B (t1 ) = G l B x2 (t)X i B (t1 − τ B )

(4.7)

The output of the adder A for t = t1 is: X a A (t1 ) = +X i A (t1 ) + X l B (t1 )

(4.8)

This is the input to the Schmitt-trigger A. The oscillator will change state (the Schmitt-trigger output will change) when X a A reaches the higher threshold level: X a A (t1 ) = Dh A

(4.9)

Using equations (4.5–4.9) we can find the value for the first time interval: T1 =

Dh A − G l B x2 (t)X i B (t0 ) − X i A (t0 ) + G l B x2 (t)K i+B τ B G l B x2 (t)K i+B + K i+A

(4.10)

Determination of T2 In the next time interval (T2 ) the Schmitt-trigger outputs have the values X o A = −1 and X oB = 1. Thus, X i A (t2 − τ A ) = X i A (t1 ) − K i−A (T2 − τ A )

(4.11)

X i B (t2 ) = X i B (t0 ) + K i+B (T1 + T2 )

(4.12)

The output of soft-limiter A, responsible for the triggering of oscillator B is modulated by signal x1 . The soft-limiter output at t = t2 is: X l A (t2 ) = G l A x1 (t)X i A (t2 − τ A )

(4.13)

The output of adder B for t = t2 is: X a B (t2 ) = X i B (t2 ) − X l A (t2 )

(4.14)

This is the input to the Schmitt-trigger block B. The oscillator will change state when X a B reaches the higher threshold level: X a B (t2 ) = Dh B

(4.15)

4.2 High Level Study

71

Using equations (4.11–4.15) the second time interval is: T2 =

Dh B + G l A x1 (t)X i A (t1 ) − K i+B T1 − X i B (t0 ) + G l A x1 (t)K i−A τ A G l A x1 (t)K i−A + K i+B

(4.16)

Determination of T3 In the third time interval (T3 ) the Schmitt-trigger outputs have the values X o A = −1 and X oB = −1. Thus, X i B (t3 − τ B ) = X i B (t2 ) − K i−B (T3 − τ B )

(4.17)

X i A (t3 ) = X i A (t1 ) − K i−A (T2 + T3 )

(4.18)

The output of the soft-limiter B is modulated by the input signal x2 . The softlimiter output at t = t3 is: X l B (t3 ) = G l B x2 (t)X i B (t3 − τ B )

(4.19)

The output of the adder A for t = t3 is: X a A (t3 ) = X i A (t3 ) + X l B (t3 )

(4.20)

This is the input to the Schmitt-trigger A. The oscillator will change state when X a A reaches the lower threshold level: X a A (t3 ) = Dl A

(4.21)

Using (4.17–4.21) the third time interval is: T3 =

−Dl A + G l B X i B (t2 )x2 (t) − K i−A T2 + X i A (t1 ) + G l B x2 (t)K i−B τ B G l B K i−B x2 (t) + K i−A

(4.22)

Determination of T4 In the fourth and last time interval (T4 ) the Schmitt-trigger outputs are X o A = 1 and X oB = −1. Thus, X i A (t4 − τ A ) = X i A (t3 ) + K i+A (T4 − τ A )

(4.23)

X i B (t4 ) = X i B (t2 ) − K i−B (T3 + T4 )

(4.24)

72

4 Quadrature Oscillator-Mixer

The soft-limiter A output is modulated by the first input signal x1 . The soft-limiter output at t = t4 is: X l A (t4 ) = G l A x1 (t)X i A (t4 − τ A )

(4.25)

The output of the adder B for t = t4 is: X a B (t4 ) = X i B (t4 ) − X l A (t4 )

(4.26)

This is the input to the Schmitt-trigger B. The oscillator will change state when X a B reaches the lower threshold level: X a B (t4 ) = Dl B

(4.27)

Using (4.23–4.27) the fourth time interval is: T4 =

−Dl B − G l A x1 (t)X i A (t3 ) − K i−B T3 + X i B (t2 ) + G l A x1 (t)K i+A τ A G l A x1 (t)K i+A + K i−B

(4.28)

Oscillation frequency and phase error We have now only 4 equations for 9 variables (T1 , T2 , T3 , T4 , X i A (t0 ), X i B (t0 ), X i A (t1 ), X i B (t2 ), X i A (t3 )). Thus, we need five additional equations, to obtain T1 , T2 , T3 and T4 . Using equations (4.5), (4.12), and (4.18) reduces the number of variables to six, but two additional equations are still necessary. The remaining two equations are obtained by assuming that the circuit is in steady-state: K i+A (T1 + T4 ) = K i−A (T2 + T3 )

(4.29)

K i+B (T1 + T2 ) = K i−B (T3 + T4 )

(4.30)

With equations (4.29–4.30) we can obtain T1 , T2 , T3 , and T4 . These can be used to calculate the duty-cycle, oscillation frequency, and phase difference relationship of the outputs. We use the following definitions: (1) Duty-cycle dc A =

T1 + T4 T

and

dc B =

T1 + T2 T

(4.31)

(2) Oscillation frequency f0 =

1 T1 + T2 + T3 + T4

(4.32)

4.2 High Level Study

73

(3) Phase difference ␾ = 180

T1 + T3 T

(4.33)

We also use the following notation Ki A =

K i+A + K i−A 2

Ki B =

⌬K i A =

K i+A − K i−A 2

⌬K i B =

D A = Dh A − Dl A

K i+B + K i−B 2

(4.34)

K i+B − K i−B 2

(4.35)

D B = Dh B − Dl B

(4.36)

R Again, the results are obtained by using a symbolic algebraic program (Derive).

  1 ⌬K i A 1− dc A = 2 Ki A ⎡

1−

  1 ⌬K i B 1− dc B = 2 Ki B

(4.37) ⎤

⌬K i A ⌬K i B Ki A Ki B

⎥ ⎢  ⎥ ⎢ 2  ⎥ ⎢ ⌬K i A ⎥ ⎢ 1 −  ⎥ ⎢  Ki A ⎥ ⎢ 2K i A DB ⎥ ⎢+ + τ A ⎥ ⎢ G 2G x (t)K D x (t)K G l A x1 (t)D A DA lB 2 iB B lB 2 iB ⎥ ◦⎢ 1 + + (τ + τ ) A B ␾ = 90 ⎢ ⎥ G l A x1 (t)K i A D A DA ⎥ ⎢ ⎥ ⎢    2 ⎥ ⎢ ⌬K i B ⎥ ⎢ 1 − ⎥ ⎢   ⎥ ⎢ K i B 2K i B DA ⎥ ⎢− + τ B ⎦ ⎣ 2G G K D x (t) x (t)K G l B x2 (t)D B DB lA iA A 1 lA 1 iA + 1+ (τ A + τ B ) G l B K i B D B x2 (t) DB (4.38)

⎡



 1−

⌬K i A Ki A

2

⎤

⎥ ⎢ ⎥ ⎢ Ki A ⎥ ⎢ ⎥ ⎢ DA G l B x2 (t)K i B D B 2G l B x2 (t)K i B ⎢ 1+ + (τ A + τ B ) ⎥ ⎥ ⎢ G l A x1 (t)K i A D A DA 1⎢ ⎥ f0 = ⎢ ⎥  2  ⎥ 2⎢ ⌬K i B ⎥ ⎢ 1− ⎥ ⎢ ⎥ ⎢ Ki B Ki B ⎥ ⎢+ ⎦ ⎣ D G l A x1 (t)K i A D A 2G l A x1 (t)K i A B 1+ + (τ A + τ B ) G l B x2 (t)K i B D B DB

(4.39)

74

4 Quadrature Oscillator-Mixer

From (4.37–4.39) we arrive at the following conclusions for the oscillator-mixer: (1) The integration constant, K i , controls the oscillator duty-cycle. The introduction of the mixing function does not affect the duty-cycle. (2) If we consider only mismatches in the integrator difference of positive and negative slopes ⌬K i A and ⌬K i B , and if the coupling gain is very high and with x1 (t) = x2 (t) = 1, equation (4.38) simplifies to   ⌬K i A ⌬K i B ␾ = 90◦ 1 − Ki A Ki B

(4.40)

which means that the quadrature error is the product of two relative errors, becoming a second order error term. (3) In order to make it easier to see the meaning of (4.38) we consider that the same modulating signal is injected in both soft-limiters x(t) = x1 (t) = x2 (t), and we make the following simplifying assumptions: (3a) If the only non-ideal effect is a mismatch between the two integration constants, K i A and K i B , 

1 Ki A − Ki B 1+ G l x(t) K i A + K i B

◦

␾ = 90

 (4.41)

(3b) If the only non-ideal effect is the mismatch of the difference between the decision values of the Schmitt-triggers, D A and D B ,  ␾ = 90◦ 1 +

1 DB − DA G l x(t) D A + D B

 (4.42)

(3c) If all blocks are symmetric and matched, but there are delays,  ◦

␾ = 90

1−

D Ki

␶ A − ␶B + G l x(t)(␶ A + ␶ B )

(4.43)

Equations (4.41–4.43) show that mismatches and delays in the circuit, which are responsible for a first order effect on the quadrature error, are attenuated by the soft-limiter gain. For example if we have a 10% deviation in one parameter, with x1 (t) = x2 (t) = 1, and the soft-limiter gain is 10, the quadrature error is about 1◦ , which is a low value. Equations (4.41–4.43) also show that the modulation signal has some influence on the phase difference. If there are mismatches and modulating signal some phasenoise is originated; by increasing the coupling gain this influence can be neglected. (4) Without any mismatches equation (4.39) can be simplified to f0 =

Ki 2D + 4G l x(t)K i ␶

(4.44)

4.3 Circuit Level Study

75

Ki If the delay is neglected (␶ = 0), f 0 = 2D , which is the ideal result obtained in Chapter 3. (5) The soft-limiter gain is a critical parameter. It should be high to minimize variations in frequency and phase. However, a high G l reduces the maximum oscillation frequency in the presence of delays, as shown by equation (4.44). Since delays are usually small, the reduction of frequency can be neglected.

4.3 Circuit Level Study In order to validate the previous high level analysis at circuit level we designed a 2.4 GHz oscillator-mixer with the circuit in Fig. 4.7. In this section we will not do a theoretical study at circuit level, but, instead we will perform simulations concerning the following parameters: quadrature relationship, gain, oscillation frequency, and phase-noise. Other mixer parameters, such as intermodulation distortion (IIP2 and IIP3 ), 1-dB compression point, input and output impedance, noise figure, and LO leakage, need further analysis. To achieve an oscillation frequency of 2.4 GHz the circuit was designed with the following parameters: R = 100 ⍀, M transistors with (W/L) = 200 ␮m/0.35 ␮m, and soft-limiter transistors M SL , with 80 ␮m/0.35 ␮m, C1 = C2 = 420 fF, I1 = I2 = 3 mA, and I SL = 1 mA, supply voltage VCC = 3 V (this is the same circuit of the example in Chapter 3). The results in Figs. 4.8 and 4.9 are obtained with i SL = 1 + 0.2 cos(␻1 t) [mA] with f 1 = 300 MHz. In Figs. 4.8 and 4.9 we represent the oscillator outputs (νOUT1 VCC

R

R

R

R

vOUT1

vOUT2

M

M

M

M

C2

C1

MSL

I1

MSL

MSL

iSL

I1

Fig. 4.7 Oscillator-mixer circuit implementation

I2

MSL

iSL

I2

76

4 Quadrature Oscillator-Mixer 0

f0 −10 −20

f0 – f1

f0 + f1

Amplitude (dBV)

−30 −40 −50 −60 −70 −80 −90

0

1

2

3

4

5

6

Frequency (GHz)

Fig. 4.8 Oscillator-mixer output in the frequency domain

0.8

vOUT1 vOUT2 0.6

0.4

Amplitude (V)

0.2

0

−0.2

−0.4

−0.6

−0.8 10

11

12

13

14

15

16

Time (ns)

Fig. 4.9 Oscillator-mixer modulated output in the time domain

17

18

19

20

4.3 Circuit Level Study

77

and νOUT2 ), in the frequency and in the time domains, which show that this crosscoupled oscillator, can be used as a combined oscillator-mixer. At circuit level, when we change the limiter bias current we change its gain and the limiting levels at the same time. The dominant effect is the changing of the limiting levels (the change in the coupling gain can be neglected). An amplitude modulated (AM) carrier with modulating signal x(t) is [58]: xc (t) = Ac [1 + ␮x(t)] cos(␻0 t)

(4.45)

where Ac is the unmodulated carrier amplitude, and ␮ is a positive constant (modulation index). In the circuit of Fig. 4.7 the amplitude of the output square-waves is Vom = 4R(I + i SL )

(4.46)

Since I = 3 mA, and i SL = 1 + 0.2 cos(␻1 t), the modulation index is ␮ = 0.05. In Figs. 4.9 and 4.10 (Fig. 4.10 is a zoom of Fig. 4.9) we can observe the two oscillator outputs in the time domain. The mixer gain is about 500 mV/mA (the input is a sinusoidal current with 0.2 mA amplitude and the output is a sinusoidal voltage with 100 mV amplitude). The modulation index is about 0.1, which is the double of the expected value. This difference can be explained by noting that, at high frequency, the waveforms are approximately sinusoidal with lower amplitude than in the theory (about half in our simulations). 0.8

vOUT2

vOUT1 0.6

0.4

Amplitude (V)

0.2

0

−0.2

−0.4

−0.6

−0.8 20

20.1

20.2

20.3

20.4

20.5

20.6

20.7

Time (ns)

Fig. 4.10 Oscillator-mixer quadrature outputs in the time domain (zoom)

20.8

20.9

21

78

4 Quadrature Oscillator-Mixer Table 4.2 Effect of 10% increase in current I1 (Fig. 4.7)

I S L (mA)

Oscillator-mixer (␮ = 0.05)

Oscillator ◦

Frequency (GHz)

Phase Difference ( )

Frequency (GHz)

Phase Difference (◦ )

1 10

2.42 2.14

91.9 90.2

2.42 2.14

92.0 90.3

I S L (mA)

Oscillator

Table 4.3 Effect of 10% decrease in current I1 (Fig. 4.7) Oscillator-mixer (␮ = 0.05) ◦

Frequency (GHz)

Phase Difference ( )

Frequency (GHz)

Phase Difference (◦ )

1 10

2.38 2.14

88.2 89.8

2.38 2.14

88.1 89.7

I S L (mA)

Oscillator

Table 4.4 Effect of 10% increase in C1 (Fig. 4.7) Oscillator-mixer (␮ = 0.05) ◦

Frequency (GHz)

Phase Difference ( )

Frequency (GHz)

Phase Difference (◦ )

1 10

2.35 2.10

85.7 89.2

2.35 2.10

85.4 88.7

I S L (mA)

Oscillator

Table 4.5 Effect of 10% decrease in C1 (Fig. 4.7)

1 10

Oscillator-mixer (␮ = 0.05) ◦

Frequency (GHz)

Phase Difference ( )

Frequency (GHz)

Phase Difference (◦ )

2.44 2.19

93.8 90.7

2.44 2.19

94.0 90.8

The influence of mismatches on the oscillator-mixer frequency and phase relationship is shown by the simulation results in Tables 4.2–4.5. In order to change the soft-limiter gain we change the bias current of the differential pair. In Tables 4.2 and 4.3 we observe the influence on the quadrature relationship of a change in the currents in one oscillator. We consider the oscillator without modulation and with modulation index 0.05 for the two cases: I SL = 1 mA and I SL = 10 mA. In Table 4.2 we consider a 10% increase in the currents in one oscillator (I1 = 3.3 mA) when the currents in the other oscillator have the nominal value (I2 = 3 mA). In Table 4.3 we consider a 10% decrease of I1 (I1 = 2.7 mA). We also consider the influence of mismatches in the capacitors (which are also responsible for the integration constant). In Table 4.4 we consider a 10% increase in C1 (C1 = 462 fF), and in Table 4.5 we consider a 10% decrease (C1 = 378 fF), while C2 remains with the nominal value (C2 = 420 fF). From Tables 4.2–4.5 we observe that the increase in the limiter gain (by increasing I SL from 1 mA to 10 mA) reduces the oscillation frequency, which is expected

4.4 Conclusions

79

–80

–90

Phase Noise (dBc/Hz)

–100

–110

Oscillator-Mixer –120

Oscillator

–130

–140

–150 105

106 107 Offset Frequency (Hz)

108

Fig. 4.11 Oscillator phase-noise with and without mixing

from (4.44), and improves the quadrature accuracy, both in the oscillator and in the oscillator-mixer, in accordance with (4.38) and (4.41). In Fig. 4.11 we plot the oscillator phase-noise with and without modulation. We observe that the mixing leads to some degradation of the oscillator phase-noise, as pointed out above; however, this degradation is negligible (in our simulation the degradation is about 1 dB).

4.4 Conclusions In this chapter we show that a quadrature relaxation oscillator can perform the mixing function if we inject a modulating signal in the oscillator feedback loop, by changing the soft-limiter gain and/or limitation levels. In a high level study, we derive equations for the duty-cycle, quadrature relationship, and oscillation frequency of the quadrature oscillator-mixer. We show that the effect of mismatches is attenuated, and becomes a second order effect. This is the main advantage of combining the oscillator and mixer, and is in contrast with the use of separate mixers, in which case mismatches are responsible for a first order quadrature error. The approach proposed here has also the important advantage of reducing the area and power consumption. A 2.4 GHz CMOS relaxation oscillator-mixer was designed with a 0.35 ␮m CMOS AMS technology to verify the theoretical study presented in [15, 65] for

80

4 Quadrature Oscillator-Mixer

the oscillator, and the study in this chapter for the oscillator-mixer. We change the value of the coupling gain by changing the limiter bias current. We show that, with high coupling gain and with 10% relative mismatches in the capacitances or in the current sources, there is a significant reduction in the quadrature errors. A high coupling gain implies a small reduction of the oscillation frequency, which can be accommodated by adjusting the oscillator tuning.

Chapter 5

Quadrature LC-Oscillator

Contents 5.1 5.2 5.3 5.4 5.5 5.6 5.7

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single LC Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quadrature LC Oscillator Without Mismatches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quadrature LC Oscillator with Mismatches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Q and Phase-Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quadrature LC Oscillator-Mixer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81 82 85 89 92 96 98

5.1 Introduction Quadrature oscillator structures with feedback have been shown to provide accurate quadrature outputs. In Chapter 3 we studied the quadrature relaxation oscillator composed of two cross-coupled RC oscillators. In this chapter we study the quadrature LC oscillator composed of two cross-coupled LC oscillators. In a relaxation oscillator the transistors act as switches, and the operation is highly nonlinear. These oscillators are known to have higher phase-noise than LC oscillators, which operate approximately linearly. Recently [11] the coupling of two LC oscillators was proposed to obtain quadrature outputs, and the study of this type of oscillators is still in its initial stage. In this chapter we study the effect of cross coupling two LC oscillators, and we evaluate the effect of mismatches on the oscillation frequency (which can be different from the free-running frequency of the individual oscillators); we also study the effect on Q and the degradation of phase-noise. This phenomenon, phase-noise degradation due to coupling, was observed independently by some of the authors of the present book and by another research group; these findings have been reported simultaneously [20, 63]. Single LC oscillators have better phase-noise performance than single relaxation oscillators; however, when LC oscillators are coupled to provide quadrature outputs, there is some phase-noise degradation. We show that a strongly coupled relaxation oscillator can have a phase-noise performance similar to that of strongly coupled LC oscillators. L.B. Oliveira et al., Analysis and Design of Quadrature Oscillators,  C Springer Science+Business Media B.V. 2008

81

82

5 Quadrature LC-Oscillator

In the previous chapter we have shown that in a cross-coupled relaxation oscillator we can incorporate the mixing function without affecting the quadrature relationship. In this chapter we show that the same combination of functions is possible in quadrature LC oscillators, with preservation of accurate quadrature.

5.2 Single LC Oscillator The basic circuit of an LC tuned oscillator and the basic model to predict phasenoise were summarized in Chapter 2. In this section we consider the noise sources and calculate the phase-noise of a single LC oscillator. This oscillator (represented in Fig. 5.1) has been of interest for a long time [67–74], and, due to its low phasenoise, it was extensively studied [75–82]. A simple equation for the tank voltage amplitude can be determined assuming that the differential pair in steady-state switches the tail current (IT ) into either branch of the LC resonator [43]. The differential pair is modelled as a current source with a current that jumps between two extreme values (assuming that the transistors are ideal switches), connected in parallel with the RLC tank. It is assumed that R p is the equivalent parallel tank resistance (Fig. 5.2). Considering the model of Fig. 5.2, the voltage harmonics are strongly attenuated by the LC tank, and at the resonance frequency the impedances of inductor and capacitor cancel, leaving only the parallel resistance. The LC tank acts as a filter [43]: the square-wave current, represented in Fig. 5.3 a, generates an approximately sinusoidal voltage across the resonator, shown in Fig. 5.3b, with amplitude:

VCC

L 2

L 2

2C

2C

M

M

IT

Fig. 5.1 LC oscillator

5.2 Single LC Oscillator

83

Fig. 5.2 Equivalent circuit of the LC oscillator i(t)

Fig. 5.3 Tank circuit. (a) Current waveform. (b) Voltage waveform

Rp

i(t)

L

C

v(t)

v(t)

Vtank

IT t

–IT a)

Vtank ≈

t

–Vtank b)

4 IT R p ␲

(5.1)

For the circuit in Fig. 5.1, if the oscillation is at very high frequency and the oscillator has low voltage amplitude, a better approximation is to assume that the differential pair is working in the linear region. In this mode of operation, referred to in the literature as current-limited, the tank current is approximately sinusoidal and the voltage amplitude is lower than (5.1), about IT R p [83]. In an LC oscillator the noise is originated in three different blocks: the lossy LC tank, the transistors of the differential pair, and the tail current source. We can calculate the phase-noise contribution of the tank assuming that the only noise source is the thermal noise [2], which is represented either as a current source across the tank with a spectral density, S(i n ) =

4kT Rp

(5.2)

or as a voltage noise source in series with the tank with spectral density S(νn ) = S(i n )|Z |2

(5.3)

where Z is the tank impedance. Using the model of Fig. 5.2, for small offset frequencies with respect to the fundamental frequency (␻m  ␻0 /2Q), it can be shown that the impedance of a LC tank can be approximated by [1, 2]   1 ␻0 2 |Z (␻0 + ␻m )|2 ≈ R 2p (5.4) 4Q 2 ␻m

84

5 Quadrature LC-Oscillator

We use the definition Q [36] ␻0 Q= 2





dA d␻



2 +

d␪ d␻

2 (5.5)

√ where A = |Z ( j␻)|, ␪ = arg |Z ( j␻)|, and ␻0 = 1/ LC is the resonance frequency. In an LC oscillator d A/d␻ = 0 [36], and for Q 0 = Q(␻0 ) ␻0 Q0 = 2

    d␪  Rp C   = Rp =  d␻  L ␻0 L ␻=␻0

(5.6)

Considering that the losses in the capacitors are much lower than those in the inductors, the resonator quality factor is determined mainly by the inductor, and the parallel resistance is obtained from the inductor quality factor [2]. Using (5.3) and (5.4), 4kT S(νn ) = Rp

   R p ␻0  2Q␻

m

2  2  ␻0  = 4kT R p  2Q␻

(5.7)

m

From (5.7) we can conclude that increasing Q leads to a reduction in the noise spectral density, when all the other parameters remain unchanged. The output noise is frequency dependent, due to the filtering action of the tank: the spectral density is inversely proportional to the square of the offset frequency. This behavior is due to the fact that the voltage frequency response of an RLC tank rolls off as 1/f to each side of the center frequency, and power is proportional to the square of voltage [67]. An important aspect is that thermal noise affects both amplitude and phase, and equation (5.7) includes their combined effect. The equipartition theorem of thermodynamics [84] states that in equilibrium, and in the absence of amplitude limiting, the amplitude-noise and phase-noise powers are equal (the noise energy is split equally into amplitude and phase) [67]. We know that oscillators have an amplitude limiting mechanism and this will remove most of the amplitude-noise. Therefore, the total noise power in the oscillator will be approximately half the noise given by (5.7) [1, 2, 34]. The phase-noise spectral density is usually divided by the carrier power, 

2kT L(⌬␻) = 10 log Pcarrier



␻0 2Q␻m

2 (5.8)

2 /R p is the carrier power. It should be noted that (5.8) is only where Pcarrier = Vtank 2 valid for the 1/ f region of the noise spectrum (Fig. 2.15). A complete equation for all the spectral oscillator regions was presented by Leeson [39].

5.3 Quadrature LC Oscillator Without Mismatches

85

In order to take into account the noise of the differential pair and of the tail current source, we can introduce a factor F in (5.8). This factor is known as the “excess noise factor in the 1/ f 2 region”. Unfortunately it is very difficult to predict this factor; it is used as a fitting parameter on measured data [39, 43]. Including this term, the equation for the oscillator phase-noise in the 1/ f 2 region is: 

2k F T L(⌬␻) = 10 log Pcarrier



␻0 2Q␻m

2 (5.9)

Note that for the 1/ f 3 region we need to add to (5.9) the contribution of the flicker noise of the active devices.

5.3 Quadrature LC Oscillator Without Mismatches The simplest and most used implementation of the LC oscillator uses transistors to generate the negative conductance as represented in Fig. 5.1. However, several variations of this implementation in order to improve the oscillator phase-noise performance, can be found in the literature [85–92] Lately, there has been some research in coupled LC oscillators and several approaches have been presented [93–97]. The implementation in Fig. 5.4, first presented in [11], couples two equal LC oscillators, expecting to inherit the good phasenoise performance of the individual oscillators. The coupling block is implemented, as in the relaxation oscillator, with a differential pair that senses the voltage at one oscillator output and injects a current in the second oscillator, in order to trigger it. A linear model of a cross-coupled LC oscillator consists of two coupled parallel RLC circuits as represented in Fig. 5.5. Without mismatches, C1 = C2 = C, L 1 = L 2 = L, and R p1 = R p2 = R p . In parallel with each tank there are negative resistances −1/gm , which cancel the losses. Two differential transconductances gmc provide the coupling, and are responsible for the quadrature outputs. In the linear model of Fig. 5.5 the loop gain is: ⎞2

⎛ ⎜ 2 ⎜ G loop (s) = −gmc ⎝

 1 + sL

sL



1 − gm + s 2 LC Rp

⎟ ⎟ ⎠

(5.10)

Using the Barkhausen criterion for the loop gain we equate (5.10) to 1 with gm = 1/R p , and we solve in order to ␻ (s = j␻). From ±j =

gmc s L 1 + s 2 LC

(5.11)

86

5 Quadrature LC-Oscillator VCC L2

L2

L2

2C

L2

2C

2C

2C

v2

v1

M

M

M

M

IT

IT

MSL

MSL

MSL

ISL1

MSL

ISL2

Fig. 5.4 Quadrature LC Oscillator

i1 1 gm

L1

C1

Rp1

v1

gmc gmc i2 1 gm

L2

C2

Fig. 5.5 Linear model of the cross-coupled LC Oscillator

Rp2

v2

5.3 Quadrature LC Oscillator Without Mismatches

87

making s = j␻ gmc ␻L 1 − ␻2 LC

(5.12)

gmc 2␻L − ␻20 = 0 2C

(5.13)

±1 =

␻2 ±

From (5.13) the oscillation frequency has two solutions: ␻osc1

gmc =+ + ␻0 2C

␻osc2

gmc =− + ␻0 2C

 1+



gmc2 L 4C

g 2L 1 + mc 4C

 (5.14)

 (5.15)

Assuming that, gmc2 L 1 4C

(5.16)

(this is always satisfied with practical values) we obtain, ␻osc1 ≈ ␻0 +

gmc 2C

(5.17)

␻osc2 ≈ ␻0 −

gmc 2C

(5.18)

From equations (5.17) and (5.18) we can observe that coupling two oscillators will produce some shift in the oscillation frequency, which is in accordance with previous publications [63, 69, 93]. To find out which of the two solutions, (5.17) or (5.18), will prevail, a study of the stability of the oscillations is required, which, to our knowledge, has not been done so far (that is an interesting topic for future research work). In the measurements (to be presented in Chapter 7) we observe that it is the lower frequency that prevails. Using the Barkhausen criterion, the phase of the loop gain is a multiple of 2␲. This means that the argument of the impedance of the resonant circuit is ␾ = +␲/2 or ␾ = −␲/2, as required for quadrature outputs. The impedance of a parallel RLC circuit with the compensating conductance −1/gm is:

88

5 Quadrature LC-Oscillator

Z (s) = s L|| =

1

1 1 ||R p || − sC gm j␻L

=

1 1 − gm + j␻C + j␻L Rp

1−

␻2 LC



+ j␻L

1 − gm Rp



(5.19)

and the argument is: 

␪ = arg[Z ] =

␲ − tan−1 2

1 − gm Rp (1 − ␻2 LC)



␻L

(5.20)

θ π 2

LC RLC

ω

ω0

Fig. 5.6 Tank impedance phase with and without compensation of losses

−

π 2

V2 − gmc V1 φ

I2

θ

gm V2

I1

gmc V2

θ

Fig. 5.7 Phasor diagram without mismatches

gm V1

V1

5.4 Quadrature LC Oscillator with Mismatches

89

In Fig. 5.6 we plot the two extreme cases, without compensation of losses and with full compensation. At the resonance, ␻0 = 1/LC, without compensation of losses ␪ = 0, but with full compensation ␪ = ±␲/2. In Fig. 5.7 the currents and the voltages in the LC oscillator are represented by a phasor diagram, where ␾ represents the phase difference between the V1 and V2 , and ␪ is the phase difference between I1 and V1 , which is  ␪ = arctan

gmc gm

 (5.21)

The circuit without mismatches is symmetric and this implies that there is perfect quadrature, ␾ = ␲/2, otherwise the voltages and currents would be different in the two oscillators, which is incompatible with the circuit symmetry.

5.4 Quadrature LC Oscillator with Mismatches Without any mismatch there is total compensation of the losses, and the argument of the impedances R//L//C//gm−1 is ±␲/2. In the case of two coupled oscillators with different resonance frequencies (␻01 and ␻02 ) we do not have full compensation of losses, and there is a quadrature error. In the linear model of Fig. 5.5 the loop gain is: ⎛ ⎜ 2 ⎜ G loop (s) = −gmc ⎝

⎞⎛  1 + s L1

⎞

⎟⎜ ⎟ s L1 s L2 ⎟⎜ ⎟    ⎠⎝ ⎠ 1 1 2 2 − gm1 + s L 1 C1 1 + s L2 − gm2 + s L 2 C2 R p1 R p2 (5.22)

If we apply the Barkhausen criterion to the loop gain in equation (5.22), we do not arrive at a simple equation for the oscillation frequency. Thus, we will use a different approach to calculate the oscillation frequency. In Fig. 5.8 we represent the oscillator phasor diagram when there are mismatches. With mismatches, there is a quadrature error ⌬␾. The Barkhausen criterion imposes that ␾1 +␾2 = ␲ (␾1 = ␲/2+⌬␾, and ␾2 = ␲/2−⌬␾). The impedance phase angles, ␪1 and ␪2 , with mismatches (represented in Fig. 5.8) can be expressed as  ␪1 = arctan

 (5.23)

gm1 V1 + gmc2 V2 cos(␾1 )

 ␪2 = arctan

gmc2 V2 sin(␾1 )

gmc1 V1 sin(␾2 ) gm2 V2 + gmc1 V1 cos(␾2 )

 (5.24)

90

5 Quadrature LC-Oscillator Δφ φ2 − V1

V2 − gmc1V1

I2

φ1

gm V2

Δ θ2 θ2 θ

I1 Δ θ1 θ

θ1 gm V1

gmc2 V2

V1

Fig. 5.8 Phasor diagram with mismatches

In the following analysis we assume that the amplitudes are equal |V1 | = |V2 |, which is valid if we have a strong non-linearity (this strong non-linearity is necessary for a stable amplitude). As we discuss in Section 5.1 the current is not sinusoidal, and it is approximately a square-wave, but the fundamental component is sinusoidal, and can be represented by a phasor. The oscillator output voltage is approximately sinusoidal, since the RLC tank acts as a filter: thus, we can represent it by a phasor. We also assume that there are no mismatches in the coupling blocks, gmc1 ≈ gmc2 ≈ gmc , and in the transconductances that compensate the losses, gm1 ≈ gm2 ≈ gm . Finally, since we are close to ␲/2, we assume that sin(␾1 ) ≈ sin(␾2 ), cos(␾1 ) ≈ − cos(␾2 ). 

gmc sin(␾1 ) ␪1 = arctan gm + gmc cos(␾1 )

 ␪2 = arctan

gmc sin(␾1 ) gm − gmc cos(␾1 )

 (5.25)

 (5.26)

Note that equation (5.25) and (5.26) result in equation (5.21) if ␾ = ␲/2. In order to calculate the quadrature error we can differentiate equations (5.25) and (5.26), to determine the ⌬␾ as a function of ⌬␪1 = ␪1 − ␪ and ⌬␪2 = ␪2 − ␪:

5.4 Quadrature LC Oscillator with Mismatches

91

   gmc gmc 2 +2 cos(␾1 ) 1+ gm gm ⌬␾ = ⌬␪1     gmc gmc 2 + cos(␾1 ) gm gm

(5.27)

   gmc gmc 2 +2 −1 − cos(␾1 ) gm gm ⌬␾ = ⌬␪2     gmc gmc 2 − cos(␾1 ) gm gm

(5.28)





Considering that we are close to the quadrature ␾ ≈ ␲/2 and cos(␾) ≈ 0, for small mismatches ⌬␾ is proportional to ⌬␪.  ⌬␾ ≈ 1 +



gm gmc

2 





⌬␪1 = − 1 +

gm gmc

2  ⌬␪2

(5.29)

Note that the above study has several approximations and is only valid for small mismatches. Equation (5.29) implies that ⌬␪1 = −⌬␪2 = ⌬␪, and allows us to calculate the quadrature error ⌬␾ due to mismatches as a function of the coupling intensity. Increasing the coupling reduces the quadrature error, which is in accordance with [63]. An RLC circuit with high parallel resistance has a high quality factor, and a small deviation from the resonance gives a significant phase variation (as shown in Fig. 5.9). This means that in coupled LC oscillators with high Q resonators, small mismatches produce a high quadrature error. This explains why the first quadrature oscillators with low Q integrated inductors had a good quadrature relationship [11], whereas more recent realizations with higher Q and similar mismatches in the tank (for example 0.5%), have a quadrature error of 2◦ or higher [23, 63, 64, 98]. Having obtained the relationship between ⌬␪ and ⌬␾, we are able to calculate the oscillation frequency with mismatches. Let us consider, for simplicity, that there is a variation of the resonance frequency of the first oscillator, ␻01 = ␻0 + ⌬␻0 , and that the resonance frequency of the other θ = arg [R //L // C ]

π 2

High Q LowQ

Fig. 5.9 Phase of the tank impedance for different Qs

−

π 2

ω0

ω

92

5 Quadrature LC-Oscillator

oscillator is not affected, ␻02 = ␻0 . The oscillation frequency changes, becoming ␻osc + ⌬␻osc , and the phase of both oscillators will change: ⌬␪1 = ␪1 − ␪ and ⌬␪2 = ␪2 − ␪. The phase of the first oscillator has a change due to mismatches and another due to the variation of the oscillator frequency, i.e., ⌬␪1 = f (⌬␻0 , ⌬␻osc ), while the phase of the second oscillator is only affected by the change on the oscillation frequency, i.e., ⌬␪2 = f (⌬␻osc ); we have the relationship ⌬␪1 = −⌬␪2 = ⌬␪. The impedance phase of an RLC circuit at ␻osc is

␪(␻osc ) =

␲ − tan−1 2

1 Rp     ␻osc 2 1− ␻0 L␻osc

(5.30)

and

␪1 (␻osc + ⌬␻osc ) =

␲ − tan−1  2

L 1 (␻osc + ⌬␻osc )  1−

 1−

(5.31)

1 R p2 2 

(5.32)

␻osc + ⌬␻osc ␻0 + ⌬␻0

L(␻osc + ⌬␻osc )

␲ ␪2 (␻osc + ⌬␻osc ) = − tan−1  2

1 R p1 2 

␻osc + ⌬␻osc ␻0

Using equations (5.30) and (5.31) we determine ⌬␪1 = ␪1 − ␪. With equations (5.30) and (5.32) we determine ⌬␪2 = ␪2 − ␪. Knowing that ⌬␪1 = −⌬␪2 , we can obtain the oscillation frequency, but this leads to complicated equations (it may be convenient to use a symbolic analysis program).

5.5 Q and Phase-Noise Equation (5.5) defines the tank quality factor. In this section we will analyse the influence on Q of deviations of the resonance frequency. The impedance phase of an RLC circuit is 1 L␻ R π p ␪(␻) = − tan−1 2 (1 − LC␻2 ) and

(5.33)

5.5 Q and Phase-Noise

93

d␪(␻) L R P (C L␻2 + 1) = d␻ C 2 L 2 R 2p ␻4 − 2C L R p 2 ␻2 + L 2 ␻2 + R p

(5.34)

Substituting (5.34) in (5.5) (and knowing that d A/d␻ ≈ 0 [36, 96]) we obtain ␻0 Q= 2

    L R P (C L␻2 + 1)    2 2 2 4   C L R p ␻ − 2C L R p 2 ␻2 + L 2 ␻2 + R p 

(5.35)

At the resonance frequency, (5.34) simplifies to: d␪(␻) = −2C R P d␻

(5.36)

and Q is given by (5.6), as expected. A maximum of 3 dB improvement of the oscillator phase-noise can exist in two coupled LC oscillators when compared with a single LC oscillator [62–64, 96]. In order to take into account this improvement, we introduce a factor 1/2 in (5.9), 

1 2k F T L{⌬␻} = 10 · log 2 Pcarrier



␻0 2Q⌬␻

2 (5.37)

This maximum 3 dB improvement, due to coupling, is obtained when the oscillators are isolated and oscillate at their common resonance frequency. When the oscillators are coupled, the oscillation frequency changes according to (5.17) or (5.18), and the theoretical equation (5.35) for Q should be used. From (5.35) and (5.37) we conclude that cross coupling two LC oscillators leads to phase-noise degradation, due to the reduction of Q (even without mismatches). To illustrate the effect of a frequency shift on Q and phase-noise we consider, as an example, an RLC circuit with a resonance frequency of 5 GHz and different values of Q 0 (value of Q at the resonance frequency): Q 0 = 5 (typical value of integrated inductors), Q 0 = 10 (typical value of high performance integrated inductors), and Q 0 = 30 (typical value for external inductors or for integrated inductors in recent technologies with special RF options). Figs. 5.10 and 5.11 show the theoretical plots of Q and of the phase-noise degradation as a function of frequency. In Figs. 5.10 and 5.11 it is clear that, with high Q inductors, as the frequency of oscillation deviates from the nominal value of 5 GHz, Q degrades steeply, and the phase-noise degrades accordingly. It can also be observed that the highest the Q, the more steeply the phase-noise degrades. The most important conclusion from this study is that the performance of single LC oscillators is different from the performance of coupled LC oscillators. Single LC oscillators oscillate at the resonance frequency and have good phase-noise performance, being the right choice for applications with very stringent phase-noise demands (e.g. GSM [99,100]). Coupled LC oscillators can have a theoretical phasenoise improvement of 3 dB, but, any deviation from the resonance frequency due

94

5 Quadrature LC-Oscillator 30

Q0 = 30 25

Quality factor

20

15

Q0 = 10 10

Q0 = 5

5

0 −1000

−800

−600

−400 −200 0 200 400 600 Offset from resonance frequency [MHz]

800

1000

Fig. 5.10 Variation of Q due to change in the oscillation frequency with respect to the resonance frequency 45

Phase-noise degradation [dB]

40

Q0 = 30

35 30 25

Q0 = 10

20 15

Q0 = 5

10 5

0

0

100

200 300 400 500 600 700 800 Offset from resonance frequency [MHz]

900

1000

Fig. 5.11 Phase-noise degradation due to change in the oscillation frequency with respect to the resonance frequency

5.5 Q and Phase-Noise

95

to mismatches and to coupling will reduce Q and increase the phase-noise; this effect is more severe for high Q resonators. Thus, for coupled LC oscillators to take advantage of Q enhancement provided by modern technologies, it is mandatory that the process mismatches are low. In order to demonstrate the study above, a quadrature LC oscillator (Fig. 5.4) was designed, for an oscillation frequency of 5 GHz, using the same 0.18 ␮m CMOS technology that was used in the example of a quadrature RC oscillator in Chapter 3. The coupled oscillators have the following circuit parameters: (W/L) = 100 ␮m/0.18 ␮m for M transistors, (W/L) = 100 ␮m/0.18 ␮m for M SL transistors, Q 0 = 10, I = 1 mA, and a supply voltage of 1.8 V. Figure 5.12 shows the simulated phase-noise for a stand-alone LC oscillator and for the cross-coupled LC oscillator. In a stand-alone LC oscillator the phase-noise is −131.65 dBc/Hz @ 10 MHz. With weak coupling the quadrature oscillator has a 3 dB improvement when compared with the single oscillator. With strong coupling the phase-noise increases to −123.90 dBc/Hz @ 10 MHz. This is due to the frequency shifts originated by the coupling, which means that the oscillators oscillate at a frequency for which the Q value of each oscillator is lower than that for the resonance frequency ␻0 . In Chapter 7 we will present a comparison of the quadrature LC oscillator in this example with the quadrature RC relaxation oscillator of the example in Chapter 3. The oscillators in the two examples have the same oscillation frequency (5 GHz) and have been designed with the same technology.

–40 1) Stand alone 2) ISL = 0.4 mA 3) ISL = 3 mA

Phase Noise [dBc/Hz]

–60

–80

–100

–120

–140

–160 104

105

106 Offset Frequency [Hz]

Fig. 5.12 Phase-noise in cross-coupled LC oscillators

107

108

96

5 Quadrature LC-Oscillator

In Chapter 7 we compare not only simulation results, but we also give measurement results which confirm that quadrature RC oscillators might be a viable alternative to quadrature LC oscillators when area and cost are to be minimized.

5.6 Quadrature LC Oscillator-Mixer In this section we will not perform a theoretical study of the LC oscillator-mixer. We will only show that it is possible to perform mixing in the LC cross-coupled oscillator and we will compare the results with the RC oscillator. To evaluate the possibility of performing the mixing function in the quadrature LC oscillator we insert an input signal in the feedback loop using the same approach used for the cross-coupled relaxation oscillator: the soft-limiter tail current (i SL ) has a modulating signal, in addition to a DC component. The circuit used to perform the simulations is the cross-coupled LC oscillator of Fig. 5.4 in AMS 0.35 ␮m CMOS technology, with a supply voltage of 2 V. The circuit parameters are (W/L) = 50 ␮m/0.35 ␮m for M transistors, (W/L) = 50 ␮m/0.35 ␮m for M SL transistors, I = 2 mA, and I SL = 1 mA. For the resonators we use L = 10 nH and C = 420 fF (including parasitics) to achieve an oscillation frequency of 2.4 GHz, with a resonator Q = 5. We have used i SL = 1 + 0.5 cos(␻1 t) [mA] with f 1 = 300 MHz. In Fig. 5.13 we observe the quadrature outputs in the time domain, modulated by the injected 0.8 v1

v2

0.6

Amplitude (V)

0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 20

21

22

23

24

Fig. 5.13 Quadrature outputs with modulation

25 26 Time (ns)

27

28

29

30

5.6 Quadrature LC Oscillator-Mixer

97

0.8 v2

v1

0.6

Amplitude (V)

0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 20

20.1

20.2

20.3

20.4

20.5

20.6

20.7

20.8

20.9

21

Time (ns)

Fig. 5.14 Quadrature modulated outputs (zoom)

–80 –90

Phase noise (dBc/Hz)

–100 Oscillator-Mixer

–110 –120

Oscillator

–130 –140 –150 –160 105

106 107 Offset frequency (Hz)

Fig. 5.15 Quadrature LC oscillator phase-noise with and without mixing

108

98

5 Quadrature LC-Oscillator

signal (with a gain of 50 mV/0.5 mA), where the modulation index is 0.05 (50 mV of modulation signal for 1 V of amplitude). In Fig. 5.13 we can observe the modulated signals, but we can not evaluate whether the oscillator outputs remain accurately in quadrature. The zoom in Fig. 5.14 shows that the oscillator outputs are in quadrature. It should be noted that the mixing results from the circuit non-linearity, and an LC oscillator is, typically, a quasi-linear circuit; furthermore, the LC tank acts as a filter that strongly attenuates any signal with frequency different from the central frequency. Thus, this LC oscillator-mixer exhibits a low modulation index, when compared with the relaxation oscillator-mixer. The phase-noise plot for the oscillator with and without modulation is shown in Fig. 5.15. We observe that the modulation leads to a slight degradation in the phasenoise for low offset frequencies. The simulated phase-noise is −135.9 dBc/Hz @ 10 MHz (this simulation is done with spectreRF considering ideal current sources). The expected result from equation (5.37) at the same offset is −134.5 dBc/Hz @ 10 MHz using F = 2, which is a typical value for this parameter. Recently some work was done concerning the exact determination of the factor F, considering the noise from the transistors and the current source [43].

5.7 Conclusions Single LC oscillators are widely used due to their good phase-noise performance. In this chapter we show that there is phase-noise degradation if two LC oscillators are coupled to obtain a quadrature oscillator. It is shown that to obtain good phasenoise performance, the technology used should provide high Q inductors and good matching. Otherwise, the good phase-noise performance of a single LC oscillator, will be lost by the effect of mismatches when two LC oscillators are coupled. We present simulations of a quadrature LC oscillator, at 5 GHz, in which we study the effect of coupling on the oscillator phase-noise. Single LC oscillators oscillate at the resonance frequency and have low phase-noise. We observe that with weak coupling (with a negligible shift of the oscillation frequency with respect to the resonant frequency of individual oscillators) there is a reduction in the oscillator phase-noise of about 3 dB. However, the deviation from the resonance frequency due to strong coupling (necessary for low quadrature error as shown in equation (5.29)), produces a significant degradation of the quadrature LC oscillator phase-noise. We show that mixing can be done in LC quadrature oscillators and that the outputs remain accurately in quadrature. However, an LC oscillator is tuned to a single frequency, and the injection of any signal at a different frequency is strongly attenuated by the LC tank, which leads to a low modulation index.

Chapter 6

Two-Integrator Oscillator

Contents 6.1 6.2

6.3 6.4 6.5 6.6

6.7

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . High Level Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Non-Linear Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Quasi-Linear Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Circuit Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phase-Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two-Integrator Oscillator-Mixer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 High Level Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Circuit Implementation and Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99 100 100 102 104 107 111 113 113 114 117

6.1 Introduction Integrated wireless systems capable of operating with different frequency bands and different telecommunications standards are of great interest. In this chapter we propose a quadrature oscillator that is able to operate in a wide range of frequencies, and which can also perform the mixing function. In previous chapters we studied the quadrature relaxation oscillator, which is strongly non-linear, and the quadrature LC oscillator, which is quasi-linear. The oscillator studied in this chapter is different from the two previous ones because it does not have coupling: it is a single oscillator with inherent quadrature outputs. Both relaxation and LC oscillators have a limited tuning range, which is below one decade (typically lower than 20%). The motivation for the study of this third type of oscillator is to achieve a wideband quadrature oscillator. The circuit described in this chapter has two-integrators in a feedback structure. It is an RC oscillator, but it is different from the circuit of Chapter 3: here, instead of the memory block (Schmitt-trigger) there is a second integrator. It is possible to have a higher oscillation frequency and wide tuning range (about a decade). We show that this oscillator has an interesting characteristic: it can have either a non linear behaviour (similar to relaxation oscillators) or a quasi-linear behaviour (typical of LC oscillators). L.B. Oliveira et al., Analysis and Design of Quadrature Oscillators,  C Springer Science+Business Media B.V. 2008

99

100

6 Two-Integrator Oscillator

In this chapter we investigate the two-integrator oscillator and its key parameters: tuning range, oscillation frequency, quadrature relationship, and phase-noise. We also consider the possibility of performing the mixing function without affecting the quadrature relationship between the oscillator outputs, with the advantages of reducing the overall area and power consumption of the RF front-end.

6.2 High Level Study As we have done in previous chapters we consider a high level model of the twointegrator oscillator, in order to clarify the basic principles involved in its operation.

6.2.1 Non-Linear Performance The oscillator in Fig. 6.1 is composed of two-integrators and two hard-limiters that implement the sign function, connected in a feedback loop. Each integrator output determines the input polarity of the other integrator [8]. The oscillation frequency is proportional to the integrators’ constant, and depends on the oscillator amplitude. The waveforms are rectangular at the hard-limiter outputs, and are triangular at the integrators outputs. The two-integrator outputs are represented, with the normalized value of 1, in Fig. 6.2. To analyse the oscillator performance we consider that in the circuit of Fig. 6.1 the integrator constants are equal and that for t = 0 the initial values of the integrator outputs are different: 1 for integrator 1 and 3 for integrator 2 (Fig. 6.3). The oscillation amplitude VOUT is the sum of the two initial values. The oscillation frequency and the amplitude are related by: f0 =

Ki 2VOUT

(6.1)

where K i is the integration constant, and VOUT is the output amplitude. With different integration constants, Fig. 6.4 shows that the amplitudes of the integrator outputs are different, but the outputs have the same frequency and are accurately in quadrature. The integrator output amplitudes are dependent on the initial conditions of twointegrators and on their integration constants:

vOUT1

1

Fig. 6.1 Two-integrator oscillator with hard-limiters

vOUT2

1

2

–1

2

6.2 High Level Study

101 vOUT1

vOUT2

Amplitude

1

Time

Fig. 6.2 Two-integrator oscillator triangular integrator outputs

vOUT1

Amplitude

4

Time

Fig. 6.3 Integrator outputs with equal integrator constants

vOUT2

102

6 Two-Integrator Oscillator

vOUT1

Amplitude

vOUT2

Time

Fig. 6.4 Integrator outputs with different integrator constants

  K i1 VOUT1 = 2 VINT1 + VINT2 K i2 

VOUT2

K i2 = 2 VINT2 + VINT1 K i1

(6.2)

 (6.3)

where VINT1 and VINT2 are the initial values. To determine the oscillation frequency we can substitute equations (6.2) or (6.3) in (6.1). The oscillation frequency is: 1

f0 = 4

VINT1 K i1

+

VINT2 K i2

!

(6.4)

In the high level model of Fig. 6.1 the amplitude and frequency are not defined by the circuit, but by the initial integrator values.

6.2.2 Quasi-Linear Performance The hard-limiters are critical blocks because it is difficult to design them for high frequencies. At very high frequencies the limiter is modeled as a linear amplifier (we assume unitary gain) with soft limiting (Fig. 6.5) [34]. Considering that the

6.2 High Level Study

103

vOUT1

vOUT2

1

1

2

2

–1

Fig. 6.5 Two-integrator oscillator with soft-limiters

oscillator amplitude does not saturate the soft-limiter, the oscillator has a linear behaviour, with sinusoidal outputs (Fig. 6.6), and the oscillator quadrature relationship is preserved. With this approach the circuit is linear, and the loop gain is: 

K1 K2 s2

 = G loop (s)

(6.5)

Using the Barkhausen criterion, the loop gain (6.5) is equal to 1. Solving (6.5) in order to ␻ we obtain the oscillation frequency: ␻0 =

"

K1 K2

Amplitude

vOUT1

Time

Fig. 6.6 Two-integrator oscillator with sinusoidal outputs

(6.6)

vOUT2

104

6 Two-Integrator Oscillator vOUT1

Amplitude

vOUT2

Time

Fig. 6.7 Effect of mismatches in the integrators of two-integrator oscillator

In this model the amplitude is defined by the soft-limiter saturation levels. Note that, as in the previous case, the oscillator changes the amplitude in the presence of mismatches in order to have accurate quadrature outputs (Fig. 6.7).

6.3 Circuit Implementation A two-integrator oscillator circuit is presented in Fig. 6.8. Each integrator is realized by a differential pair (transistors M) and a capacitor (C). The oscillator frequency is controlled by Itune . There is an additional differential pair (transistors M L ), with the output cross-coupled to the inputs, which performs two related functions:

r r

compensation of the losses due to R to make the oscillation possible (a negative resistance is created in parallel with C); amplitude stabilization, due to the non-linearity (the current source Ilevel controls the amplitude);

It should be noted that the correspondence between the circuit of Fig. 6.8 and the block diagrams in the previous section (Figs. 6.1 and 6.5) is conceptual and not topological: the integrators with limited output shown in Fig. 6.8 are modeled by the ideal integrators in cascade with limiters. The circuit of Fig. 6.8 can be represented by the linear model in Fig. 6.9, where the negative resistance is realized by the cross-coupled differential pair (M L ), and R represents the integrator losses due to the pairs of resistances R/2.

6.3 Circuit Implementation

105 VCC

R/2

R/2

R/2

R/2

vOUT1

ML

vOUT2

ML

ML

ML

Ilevel

M

C1

Ilevel

M

C2

M

Itune

M

Itune

Fig. 6.8 Two-integrator oscillator implementation

From the model in Fig. 6.9, which is valid for quasi-linear performance, we can obtain the oscillator frequency using the loop gain of the oscillator. For oscillation, the losses must be compensated (R p = 1/gm L ), each stage is a perfect integrator, and the phase condition is achieved for all frequencies, because each stage of the two-integrator oscillator gives a 90◦ phase shift, as required for quadrature outputs. Thus, the two-integrator oscillation frequency is determined by the amplitude condition. The loop gain is: |H ( j␻)| =

gm2 ␻2 C 2

(6.7)

C

R

1 –g

mL

–gm gm

Fig. 6.9 Two-integrator oscillator linear model

C

R

1 –g

mL

106

6 Two-Integrator Oscillator

and using the amplitude condition, |H ( j␻)| = 1, the oscillation frequency is: ␻0 =

gm C

(6.8)

From equation (6.8) we can conclude that the oscillator frequency varies by changing either the capacitance or the transconductance. In a practical circuit we can use varactors to change the capacitance or, most commonly, we can change the tuning current and therefore the transconductance. With the second approach, if the transconductances are implemented by bipolar transistors the frequency changes linearly with the tail current. If the transconductances are implemented with MOS transistors the frequency will be proportional to the square root of the tail current. Since we can change the transconductance in a wide range, these oscillations have wide tuning range. The circuit of Fig. 6.8 can work in two different modes: (1) If we over-compensate the losses by increasing Ilevel , the performance is nonlinear and resembles that of the block diagram in Fig. 6.1. With a strong non-linear performance (the transistors operate as switches) the waveforms are approximately triangular. In this case the oscillator amplitude is: VOUT ∼ = Ilevel R

(6.9)

and using (6.1) we obtain the oscillator frequency: f0 =

Itune 2C VOUT

(6.10)

In this case the oscillator has a behaviour similar to that of a relaxation oscillator. (2) If we compensate the losses only to the amount necessary for the oscillations to start, the circuit of Fig. 6.8 is modeled by that of Fig. 6.9. The transistors work in the linear region, and the outputs are close to sinusoidal with the amplitude that satisfies the condition, 1 gm L

=R

(6.11)

Since linear operation has been assumed, the currents in the transistors of the differential pair do not reach the value of the source current Ilevel (Fig. 6.10). However, we have found that in practice the output amplitude can be approximated as

VOUT ∼ = Ilevel R

(6.12)

6.4 Phase-Noise

107

Fig. 6.10 Differential voltage to current transfer characteristic of a differential pair

iD1 – iD2

Itail

v1 – v2

−Itail

6.4 Phase-Noise In this section we consider that the oscillator has a quasi-linear behaviour (sinusoidal outputs), and we calculate the oscillator phase-noise using the approach for linear oscillators [36]. In the following derivation we consider the oscillator as a feedback system, and we represent the effect of all noise sources by one equivalent noise source at the input, as shown in Fig. 6.11. The objective of the following analysis is to calculate the noise transfer function of the oscillator (Leeson’s equation). The closed-loop transfer function is N (s) =

H (s) Yn (s) = X n (s) 1 − H (s)

(6.13)

Considering frequencies ␻ = ␻0 + ␻m in the vicinity the oscillation frequency ␻0 , and using the first two terms of the Taylor expansion, the open-loop transfer function H ( j␻) is approximated by, H ( j␻) ≈ H ( j␻0 ) + ␻m

 d H  d␻ ␻=␻0

(6.14)

where H ( j␻0 ) = 1 (Barkhausen oscillation condition). Replacing (6.14) in (6.13), and using the simplified notation N [ j(␻0 + ␻m )] = N (␻m ) Noise source xn

Fig. 6.11 Linear oscillator with noise input

H(s)

yn

108

6 Two-Integrator Oscillator

N (␻m ) =

1 + ␻m dd␻H

(6.15)

−␻m dd␻H

In practical cases |␻m d H/d␻| << 1 [1, 36], so N (␻m ) ≈

−1 ␻m dd␻H

(6.16)

and the noise power spectral density at ␻ = ␻0 + ␻m is S(yn ) = |N (␻m )|2 S(xn )

(6.17)

1 # d H $2

(6.18)

with |N (␻m )|2 =

(␻m

)2

d␻

Equation (6.18) shows that at ␻ = ␻0 + ␻m the output noise spectral density, is multiplied by −(␻m d H/d␻)−2 , as shown in Fig. 6.12. We will now use the oscillator quality factor in the noise equation. The oscillator quality factor is defined as [36]: ␻0 Q= 2





dA d␻



2 +

d␪ d␻

2 (6.19)

where, A = |H ( j␻)| and ␪ = arg(H ( j␻)). Expressing H ( j␻) in the polar form, H ( j␻) = A( j␻) exp[ j␪(␻)], dH = d␻



 dA d␪ + jA exp( j␪) d␻ d␻

N (ω)

(6.20)

Output noise spectral density

2

Input noise spectral density

Fig. 6.12 Oscillator phase-noise [36]

ω

ω0

ω

ω0

ω

6.4 Phase-Noise

109

Equation (6.18) can be rewritten as, |N (␻m )|2 ≈

(␻m )2

%#

1

$ dA 2 d␻

+ A2

# d␪ $2 &

(6.21)

d␻

Noting that A ≈ 1 in steady-state oscillations, from (6.19) and (6.21), we obtain 1 |N (␻m )| ≈ 4Q 2



2

␻0 ␻m

 (6.22)

This equation is known as “Leeson’s equation” [39]. In a two-integrator oscillator H ( j␻) is H ( j␻) =

␻20 ␻2

(6.23)

and d␪ =0 d␻

(6.24)

␻2 dA = −2 03 d␻ ␻

(6.25)

At the oscillation frequency equation (6.25) simplifies to:  d A  2 =−  d␻ ␻=␻0 ␻0

(6.26)

and from (6.19) the two-integrator oscillator quality factor at the oscillation frequency is: ␻0 Q0 = 2





2 ␻0

2 + (0)2 = 1

(6.27)

The value of Q is unity, which is lower than the typical values for an LC oscillator. This explains the poor performance in terms of phase-noise of the two-integrator oscillator when compared with the LC oscillator. We consider now that all noise sources are represented by two independent current noise sources i n1 and i n2 as shown in Fig. 6.13.

110

6 Two-Integrator Oscillator

Fig. 6.13 Oscillator with noise current sources

ino in1

−

R

1 gmL1

C

−gm gm

−

C

1 gmL2

R

in2

Using (6.23) and (6.13) the noise response for a noise current i n1 is N1 (␻m ) =

␻20 ␻2

1−

(6.28)

␻20 ␻2

Using the approximation of (6.14) ␻20 2 ≈ 1 − ␻m ␻2 ␻0

(6.29)

and (6.28) becomes N1 (␻m ) ≈

1 ␻0 =− 2 2␻m 1 − 1 − ␻m ␻0

(6.30)

It can easily be shown that in the circuit of Fig. 6.13 the two noise sources i n1 and i n2 have the same effect on i no if we are close to the resonance frequency. Using this approximation, the total noise power density, due to i n1 and i n2 , is S[i no (␻m )] =

1 4



␻0 ␻m

2 [S(i n1 ) + S(i n2 )]

(6.31)

where S(i n1 ) = S(i n2 ) = 4kT /R, if we consider only the thermal noise of resistors R. Equation (6.31) includes both amplitude and phase-noise, and since L(␻m ) only takes into account the phase-noise an extra factor 12 [67] should be included. Using equation (6.31) the two-integrator oscillator phase-noise is  L(␻m ) = 10 log

1 S[i n out (␻m )] 2 2 Irms





4kT = 10 log 2 R Irms



␻0 2␻m

2 (6.32)

6.5 Simulation Results

111

where Irms is the rms current at the output of transconductance gm in Fig. 6.13. It should be noted that (6.32) is only valid for the 1/ f 2 region of Fig. 2.15; for the 1/ f 3 region we need to add to (6.32) the contribution of the flicker noise of the active devices.

6.5 Simulation Results The circuit of Fig. 6.8 was designed for the 0.35 ␮m technology of AMS. In Fig. 6.14 a plot of the circuit layout is shown. All the simulations are done with spectreRF, using RF models for the circuit components. The total circuit area is 500 × 400 ␮m2 , of which 150 × 100 ␮m2 corresponds to the oscillator (the circuit area is dominated by the pads). The circuit parameters are: R = 400 ⍀, (W/L) = 160 ␮m/0.35 ␮m for M transistors, (W/L) = 80 ␮m/0.35 ␮m for M L transistors, C ≈ 200 fF, and Ilevel = 2 mA. The supply voltage is 3 V. We obtain an extended tuning range, 900 MHz to 5.8 GHz, by changing the tuning current, Itune . The circuit outputs are close to sinusoidal (one of the two cases in the high level study). The frequency is proportional to the transconductance of the MOS differential pair, which, in turn, is proportional to the square root of the current. This explains the shape of the curve of frequency versus coupling current in Fig. 6.15. Figure 6.16 shows that if we change the integrator constants by 10%, the oscillator amplitudes change, but the oscillator outputs remain in quadrature (the quadrature error is 2.1◦ ). This is in accordance with the theoretical analysis (Fig. 6.7). We obtain by simulation the phase-noise for different oscillation frequencies (Table 6.1), and we compare the results with those predicted by equation (6.32) in which Ir ms is obtained by simulation. In Table 6.1, these results are obtained with

Fig. 6.14 Circuit layout

112

6 Two-Integrator Oscillator 7

Fig. 6.15 Frequency vs tuning current Frequency (GHz)

6 5 4 3 2 1 0 0

1

2

3

4

5 6 Itune (mA)

7

8

9

10

0.8 vOUT2

0.6

vOUT1

Amplitude (V)

0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 16

17

18

19 20 Time (ns)

21

22

23

Fig. 6.16 Effect of changing the integrator constants by 10%

Table 6.1 Oscillator phase-noise at an offset of 10 MHz Tuning current

Ir ms

Frequency

Eq. (6.32)

Simulated

836 ␮A 1.658 mA 1.87 mA 2.19 mA 8.27 mA

580 ␮A 1.20 mA 1.30 mA 1.55 mA 3.70 mA

900 MHz 1.8 GHz 2.0 GHz 2.4 GHz 5.8 GHz

−120.14 dBc/Hz −120.31 dBc/Hz −120.09 dBc/Hz −120.03 dBc/Hz −119.93 dBc/Hz

−118.98 dBc/Hz −121.62 dBc/Hz −121.53 dBc/Hz −121.07 dBc/Hz −117.50 dBc/Hz

6.6 Two-Integrator Oscillator-Mixer

113

R = 400 ⍀, k = 1.38 × 10−23 (J/K) and T = 300 K, and with the Ir ms values (at the transconductance output) indicated in the table. In Table 6.1 we observe that the oscillator phase-noise is approximately constant over the tuning range and has worse values at the lower and upper ends. We observe that there is a good agreement between the simulation results and those obtained using equation 6.32. Although this is a quasi-linear oscillator, the phase-noise is similar to that of a relaxation oscillator and higher than the phase-noise of an LC oscillator. This is due to the higher quality factor of the LC oscillator, which is typically higher than 1, while the two-integrator oscillator has a quality factor of 1, as shown above.

6.6 Two-Integrator Oscillator-Mixer 6.6.1 High Level Study The modulating signals can be injected either before or after the integrator blocks. Injecting the signal before an integrator will originate FM modulation; therefore we inject the modulation signal after the integrator. In Fig. 6.17 the block diagram of the two-integrator oscillator-mixer is presented. The oscillator signals remain in quadrature after the injection of a modulating signal and the mixing function is performed. The modulating signal must be always positive, which means that it must have a DC component. In the case of sinusoidal modulating signal, we would have

v1 = v2 = A + a cos(␻1 t)

(6.33)

where |a| < A. In Fig. 6.18 a high level simulation of the oscillator-mixer is presented, in which the modulating signal has amplitude 0.3 V, with a DC component of 1 V, and frequency f 1 = 0.1 f 0 .

v1

1

v2

vOUT1

1

2

–1

Fig. 6.17 Two-integrator oscillator-mixer block diagram

vOUT2

2

114

6 Two-Integrator Oscillator 1.5

1

Amplitude (V)

0.5

0

−0.5

–1

−1.5

0

5

10

15

20

25

30

35

40

45

50

Time (ns)

Fig. 6.18 Two-integrator oscillator-mixer waveforms

6.6.2 Circuit Implementation and Simulations An oscillator-mixer was designed with the circuit of Fig. 6.8, and we apply the modulating signal to the current sources Ilevel , since we want to modulate the output amplitude (modulation of Itune would change the oscillation frequency, which might be a good solution to produce frequency modulation). It should be noted that there is not a complete correspondence between the circuit and the high level block diagram. At circuit level the amplitude stabilization based on a negative gm L (which is not present in the high level model) is used for the modulation. In spite of this difference, we believe that the study in terms of the high-level model is useful to highlight some of the basic concepts involved. The results in Figs. 6.19, 6.20, and 6.21 are obtained with ilevel = 2+0.5 cos(␻1 t) [mA] with f 1 = 300 MHz. In Figs. 6.20 and 6.21 we represent the oscillator outputs vOUT1 and vOUT2 , in the time and in the frequency domains, which confirm that this two-integrator oscillator can be used as a combined oscillator-mixer. The output voltage is vOUT ∼ = R Ilevel (1 + x(t)) cos(␻0 t)

(6.34)

where x(t) is the modulating signal. Equation (6.34) shows that we have a linear modulation. The modulation index is

6.6 Two-Integrator Oscillator-Mixer

115

0.8 vOUT1

vOUT2

0.6

Amplitude (V)

0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 10

11

12

13

14

15 16 Time (ns)

17

18

10.7

10.8

19

20

Fig. 6.19 Oscillator-mixer output in the time domain

0.8 vOUT1

vOUT2

0.6

Amplitude (V)

0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 10

10.1

10.2

10.3

10.4 10.5 10.6 Time (ns)

Fig. 6.20 Oscillator-mixer quadrature output in the time domain (zoom)

10.9

11

116

6 Two-Integrator Oscillator 0 f0 −10 f0 – f1

f0 + f1

Amplitude (dBV)

−20

−30

−40

−50

−60

−70

0

1

2

3 4 Frequency (GHz)

5

6

Fig. 6.21 Oscillator-mixer quadrature output in the frequency domain

–70 –80

Phase Noise (dBc/Hz)

–90 Oscillator-Mixer –100 Oscillator –110 –120 –130 –140 –150 105

106 107 Offset Frequency (Hz)

Fig. 6.22 Two-integrator oscillator phase-noise with and without mixing

108

6.7 Conclusions

117

␮=

|xmax | Ilevel

(6.35)

The simulated modulation index (Fig. 6.19) is about 0.22 (200 mV/900 mV), which is in accordance with (6.35). Comparing this result with those obtained for the previous relaxation and LC oscillators, we can conclude that the present circuit has the highest modulation index. The other mixer characteristics, noise factor, intermodulation distortion (IIP2 and IIP3 ), 1 dB compression point, and isolation between the two ports (LO leakage), are not discussed here. The oscillator phase-noise with and without modulating signal, represented in Fig. 6.22, shows that the influence of the mixing in the oscillator phase-noise can be neglected, as it happened in the RC and LC oscillators considered in previous chapters.

6.7 Conclusions In this chapter we study a third type of quadrature oscillator: the two-integrator oscillator. The main advantage of this oscillator, when compared with relaxation and LC oscillators, is its wide tuning range, which in a practical implementation (GHz range) can be about one decade. We presented a high-level study of a two-integrator oscillator in which we consider hard-limiting (triangular outputs) and soft-limiting (sinusoidal outputs). In both cases, mismatches cause a change of amplitude and oscillation frequency, but the outputs remain in quadrature. A circuit implementation was presented and an equation has been derived for the oscillator phase-noise. The circuit has a wide tuning range, from 900 MHz to 5.8 GHz, and is suitable for use in different applications. The circuit can work with a varying degree of non-linearity: if we compensate the losses significantly more than it is necessary for oscillation, there is a strong non-linearity, and the behaviour is close to that of a relaxation oscillator, with triangular outputs. In our design the compensation of the losses is near the minimum required for oscillation, and the oscillator behaviour is approximately linear, close to a second order oscillator with sinusoidal outputs. This two-integrator oscillator can perform the mixing function throughout its wide tuning range, while preserving the quadrature relationship. This oscillatormixer has the highest modulation index when compared with circuits considered in previous chapters.

Chapter 7

Measurement Results

Contents 7.1 7.2

7.3

7.4

7.5 7.6

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quadrature Relaxation Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Circuit Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Measurement Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quadrature LC Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Circuit Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Measurement Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quadrature Oscillator-Mixer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Circuit Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Measurement Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of Quadrature LC and RC Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

119 120 120 121 123 123 126 127 127 129 132 135

7.1 Introduction In this chapter we present measurement results of five prototype circuits, in the GHz range of frequencies, to validate the theoretical analysis concerning the quadrature error and phase-noise, and to validate the oscillator-mixer concept. We designed a quadrature relaxation oscillator at 2.4 GHz, and we confirm by measurements that strong coupling leads to a reduction of the phase-noise and decreases the quadrature error. We also designed a quadrature LC Oscillator at 1 GHz, and we confirm that, although single LC oscillators have low phase-noise, with strong coupling there is an increase of the phase-noise. The objective of the third circuit prototype is to validate the oscillator-mixer concept at 5 GHz. Two prototypes at 5 GHz are used in a comparative study of cross-coupled RC and LC oscillators. In this chapter we describe only the circuit schematics, and we present the measurement results. The complete measurement setup and the procedure used to measure the oscillator phase-noise and the quadrature error are described in appendix A. L.B. Oliveira et al., Analysis and Design of Quadrature Oscillators,  C Springer Science+Business Media B.V. 2008

119

120

7 Measurement Results

7.2 Quadrature Relaxation Oscillator In this section we present the quadrature relaxation oscillator with an oscillation frequency of 2.4 GHz. We present measurement results to show the influence of the coupling gain on the quadrature relationship and phase-noise.

7.2.1 Circuit Schematic The quadrature relaxation oscillator has been implemented using a cross-coupled topology [8], shown in Fig. 7.1. The M transistors have (W/L) = 200 ␮m/0.35 ␮m, the soft-limiter transistors, M SL have (W/L) = 70 ␮m/0.35 ␮m, C = 400 fF, transistors M1 = M2 have (W/L) = 30 ␮m/1 ␮m, and the bias currents are I = 3 mA and I SL = 2 mA. The supply voltage is 3 V.

VCC

R

R

M

M

R

R

M

M

C

C

MSL

MSL

MSL

MSL

ISL

M2

M2

M2

I

M1

M1

M1

Fig. 7.1 Quadrature relaxation oscillator schematic

M1

M1

7.2 Quadrature Relaxation Oscillator

121

Fig. 7.2 Die photo of the quadrature RC oscillator

The circuit is implemented following general good layout practices [101–103], and detailed information on the AMS CMOS 0.35 ␮m process [104]. The circuit layout must be as symmetrical as possible, and signal paths must have the same length to avoid quadrature errors. The final circuit, represented in Fig. 7.2, has a total chip area of 550 × 500 ␮m2 . The circuit area without pads is 300 × 350 ␮m2 .

7.2.2 Measurement Results Since a high precision quadrature relationship is expected, visual inspection of the waveforms does not suffice. The phase difference between two sinusoidal signals x and y can be determined by using the following procedure. If x(t) = A x sin(␻0 t + ␪)

(7.1)

y(t) = A y cos(␻0 t)

(7.2)

where ␪ represents the error in the quadrature relationship, then 2 A x A y nT

t0+nT

x(t)y(t)dt = sin(␪) ≈ ␪ t0

(7.3)

122

7 Measurement Results

where T = 2␲/␻0 is the period and n is the number of periods. This method of determining the quadrature error is robust with respect to additive noise. As we explain in Chapter 2, the image rejection ratio (IRR) in the absence of gain mismatch is [1] I RR =

␪2 4

(7.4)

The two quadrature outputs at 2.4 GHz, which have about 120 mVpp, are represented in Fig. 7.3. To have the clean plot of Fig. 7.3 we have stored 16 samples of the signal, which are averaged to remove the oscillator noise. We stored the output waveforms truncated in such a way that an integer number of periods remain. After that, any offset was removed, and the amplitude was normalized to 1. Finally, the above procedure was used to obtain the quadrature error. The measured quadrature errors were lower than 1◦ with strong coupling. By interchanging the output paths, length errors can be distinguished from circuit errors. If we reduce the coupling gain by changing the current to a weak value, the measured error is higher than 4◦ . The oscillator frequency changes between 2.47 GHz, with weak coupling, to 2.29 GHz, with strong coupling. Thus, by increasing the coupling gain, the accuracy in the quadrature relationship is improved, but the oscillator frequency is reduced. The results are summarized in Table 7.1. To measure the oscillator phase-noise we use a spectrum analyzer. Since the phase-noise has units of dB with the respect to the carrier (dBc/Hz) we calculate the noise power in a 1 Hz bandwidth and divide the result by the carrier power (in

Fig. 7.3 Oscillator quadrature outputs

7.3 Quadrature LC Oscillator

123

Table 7.1 Effect of coupling on the quadrature error for the oscillator in Fig. 7.1 I S L [mA]

Frequency [GHz]

Quadrature Error [◦ ]

1.17 (weak coupling) 1.92 3.12 (strong coupling)

2.47 2.4 2.29

4.3 2.2 0.8

Table 7.2 Effect of coupling on the phase-noise for the oscillator in Fig. 7.1 Offset frequency

Weak coupling

Strong coupling

@600 kHz @1 MHz

−93 dBc/Hz −97 dBc/Hz

−100 dBc/Hz −105 dBc/Hz

this case −8 dBm). Since the spectrum analyzer bandwidth used in the measurement of the noise power is 20 kHz and 10 log10 (20000) = 43 dB

(7.5)

to have the phase-noise in dBc/Hz we need to subtract 35 dB (8 dB for the carrier and −43 dB due to the spectrum analyzer bandwidth) from the values measured in the spectrum analyzer. In Table 7.2 the phase-noise is compared at two different frequency offsets from the carrier. The improvement of phase-noise with strong coupling is about 7 dB for both frequency offsets. It is very difficult to measure accurately the absolute value of the oscillator phase-noise. The direct measurement with the spectrum analyzer has an accuracy of ±4 dB. To increase the accuracy we should use a phase-noise measuring system. However, the relative values are not strongly affected by the inaccuracy of the measurement setup. We can conclude that to have a cross-coupled relaxation oscillator with low phase-noise and low phase error we should have a strong coupling gain, which confirms the study presented in Chapter 3.

7.3 Quadrature LC Oscillator In this section we present the 1 GHz quadrature LC oscillator and the measurement results for quadrature error and phase-noise.

7.3.1 Circuit Schematic The quadrature oscillator has been implemented using a cross-coupled LC oscillator topology [11], shown in Fig. 7.4. The implementation uses M and M SL transistors with (W/L) = 100 ␮m/0.35 ␮m, and M1 = M2 transistors with (W/L) = 30 ␮m/1 ␮m. The total capacitance value, including the parasitics from transistors and bonding pads is C ≈ 2 pF, and the

124

7 Measurement Results VCC

L C

IT

L

C

C

M

L C

M

M

M

M1

M1

ISL

L

MSL

MSL

M2

MSL

MSL

M2

Fig. 7.4 Quadrature LC oscillator schematic

total bias current is 4 mA (2 mA for each oscillator). The oscillator outputs are connected to pads, and the inductance of the external inductor plus the bonding wire inductance is 9 nH. The circuit uses a 3 V supply. The overall area without pads is 300 × 450 ␮m2 . In this design we use an external inductor. Integrated inductors have some important drawbacks: they occupy a large area, and have either very low quality factor, or need special RF options (in order to have reasonable quality factor). Since the objective is to investigate the variation of phase-noise due to coupling, we use an external inductor because it is simpler and has a much higher quality factor, necessary to put in evidence significant phase-noise degradation. The external inductor used is a SMD inductor (manufactured by Coilcraft). We choose the 0805 size for facility of soldering this type of components. The 0805 HQ series has the highest quality factors, and these ceramic chip inductors have 5% inductance tolerance and batch consistency. We use a 0805 HQ inductor with 6.2 nH @ 250 MHz, with a quality factor of 88 at 1 GHz. The model of the inductor is shown in Fig. 7.5. The values of R1 , R2 , C1 , and L are listed for each component type. We use a 0805HQ-6N2 circuit with R1 = 8 ⍀, R2 = 0.04 ⍀, C = 0.056 pF, and L = 6.4 nH.

7.3 Quadrature LC Oscillator

125

Fig. 7.5 External inductor model

R2

C1

R1

Rvar

L

The value of the frequency dependent variable resistor Rvar , depends on the skin effect and is calculated by: " (7.6) Rvar = k f where f is the frequency in Hz; k is a constant with the value of 1.15 × 10−05 for this particular case. This inductor has a self resonance frequency of 4.75 GHz, but, typically the self resonance frequency of the component model will be higher than the value measured on a circuit board, since the parasitic elements of the circuit board will lower the resonance frequency, especially for very small inductance values. The total value of the inductance includes the contribution of the wirebonding. In Table 7.3 the characteristics of a typical bonding wire with a length of 2 mm are shown [105]. We use a wire with 1.0 mil1 diameter (values in bold in Table 7.3). The length is about 3 mm, which leads to an inductance of around 3 nH. The bond pitch in the circuit is higher than in Table 7.3, leading to a much lower mutual inductance. Adding the contributions of the external inductor and the wirebonding leads to a total inductance of about 9 nH. Again the circuit is implemented following general good layout practice [101–103], using detailed information on the AMS CMOS 0.35 ␮m process [104]. The final circuit, represented in Fig. 7.6, has a total chip area of 550 × 700 ␮m2 . As in any quadrature oscillator, the circuit layout must be as symmetrical as possible, and signal paths need to have the same length to avoid quadrature errors. Table 7.3 Bonding parameters of bond wire with 2 mm (from [105]), for 90 ␮m bond pad pitch and 180 ␮m bond finger pitch Bonding wire diameter

0.8 mil

1.0 mil

1.2 mil

Resistance (⍀) Inductance (nH) Capacitance (pF) Mutual inductance (nH)

0.154 2.089 0.104 0.9798

0.103 1.996 0.122 0.9787

0.079 1.915 0.140 0.9770

1

1 mil (unit of length equal to one thousandth of an inch) is 0.0254 mm

126

7 Measurement Results

Fig. 7.6 Die photo of the quadrature LC oscillator

7.3.2 Measurement Results The oscillator outputs at 1.1 GHz, represented in Fig. 7.7, have amplitude of about 250 mVpp. To have the clean plot of Fig. 7.7, we average 16 sampled waveforms to suppress the oscillator noise. We store the output waveforms, truncated to an integer number of periods. After removing any offset the amplitude is normalized to 1. Finally, the quadrature relationship is obtained by using equation (7.3). The measured quadrature errors are about 5.6◦ over the tuning range (by interchanging the outputs we remove the path length errors). If we increase the coupling gain (by increasing the bias current) the measured error is about 2.7◦ . Therefore, a high coupling gain is necessary for the oscillator to have a lower quadrature error. To measure the oscillator phase-noise we use the spectrum analyzer. The carrier power is −2 dBm, and the spectrum analyzer bandwidth is 20 kHz. To obtain the phase-noise in dBc/Hz we need to subtract 41 dB (2–43 dB) to the values measured by the spectrum analyzer. In Table 7.4 the oscillator phase-noise for two different offsets is given. With strong coupling there is a degradation of more than 10 dB, and the oscillator frequency is reduced from 1.157 GHz to 1.051 GHz. This result is very important, because it confirms our theoretical analysis, which shows that the low phase-noise of single LC oscillators is not maintained for strongly coupled LC oscillators.

7.4 Quadrature Oscillator-Mixer

127

Fig. 7.7 LC oscillator quadrature outputs Table 7.4 Effect of coupling on the phase-noise Offset frequency

Weak coupling

Strong coupling

@600 kHz @1 MHz

−124 dBc/Hz −130 dBc/Hz

−110 dBc/Hz −117 dBc/Hz

Since we use a spectrum analyzer to measure the oscillator phase-noise, there can be a significant error in the absolute value of the phase-noise. However, the relative values are correct, which is the main objective of this measurement. We conclude that in cross-coupled LC oscillators there is an important trade-off: the coupling gain should be high to ensure good quadrature, but with high coupling gain there is degradation of the oscillator phase-noise.

7.4 Quadrature Oscillator-Mixer In this section we present the measurement results of a 5 GHz quadrature relaxation oscillator-mixer. The main objective is to validate the oscillator-mixer concept. We also compare the phase-noise performance with that of other state-of-the art oscillators.

7.4.1 Circuit Schematic We designed the oscillator-mixer together with an LNA, filters, and a PLL (this could be part of the front-end of a double conversion receiver with low IF). Only

128

7 Measurement Results

Fig. 7.8 Oscillator-mixer schematic

VCC R

R

Q1

Q1

I bias

R

C

R

Q1

Q1

C1

C1

QSL QSL

QSL QSL

LNAin1

LNAin 2

Q

Q

R

Q

Q

R

R

R

Q R

the oscillator-mixer is within the scope of this book, so the other blocks, are only discussed in appendix A. The oscillator-mixer has been implemented using the cross-coupled relaxation oscillator topology [8] shown in Fig. 7.8. The tail currents of the limiters in the cross coupling paths are supplied by an LNA, so the gain of these limiters is modulated by the LNA signal. The implementation uses Q, Q 1 , and Q SL transistors with emitter areas of 0.32 ␮m × 6 ␮m, 0.32 ␮m × 10 ␮m, and 0.32 ␮m × 6 ␮m, respectively, R = 50 ⍀, C1 = 300 fF, C = 2 pF, and the bias currents are 2 mA for the oscillator core

LNA Div2

Bias OscMix

Low Pass Filters Fig. 7.9 Die photo of the oscillator-mixer

7.4 Quadrature Oscillator-Mixer

129

transistors and 2 mA for the tail currents of the soft-limiters. The input marked Ibias in Fig. 7.8 is connected to a pad, and is used to tune the oscillator. The tuning range is around 20% f osc (1 GHz). All the transistors were dimensioned and biased to have the highest f T possible in the process. The circuit uses a 2.5 V supply. The implemented circuit has an overall area of 250 × 100 ␮m2 . The complete circuit was implemented following general layout good practice rules [101–103] and detailed information on the IBM BiCMOS6HP SiGe process [106]. The final circuit, represented in Fig. 7.9, has a total chip area of 2.0×2.4 mm2 (250 × 100 ␮m2 for the oscillator-mixer). Again, we design the circuit layout as symmetrical as possible, and signal paths have the same length to avoid quadrature errors.

7.4.2 Measurement Results The first measurement result concerns the oscillator-mixer tuning range. From the simulations it was expected that the oscillation frequency would be between 5 and 6 GHz. The measured frequency, between 4.28 GHz and 5.28 GHz, is lower by 10%. This difference can be explained by last minute changes in the circuit layout to fulfil requirements of the particular chip fabrication run used (mainly changes in the available capacitors). The measured tuning range agrees with the simulated value of about 1 GHz. For the validation of the oscillator-mixer concept we include this circuit in a PLL, as described in appendix A. The oscillator is adjusted for 5 GHz, and the available output is the output of the frequency divider by two, inside the PLL, and not the 5 GHz oscillator output. In Fig. 7.10 the spectrum of the divider output at 2.5 GHz is shown, which is in accordance with simulations. The LNA has a single-ended input and two equal current outputs, which control the two soft-limiters in the quadrature RC oscillator. We inject a signal with amplitude −30 dBm and 5.010 GHz at the LNA input; this frequency has 10 MHz difference form the oscillation frequency, which is the lowest possible frequency for a Low-IF in these 5–6 GHz bands, in which the channels have 10 MHz bandwidth. We expected an output signal of about −30 dBm, leading to an overall gain of 0 dB. However, the flip-chip gold pads used in the simulation, were later found not to be available in the particular chip fabrication run used, leading to last minute use of wirebond pads in the circuit layout. The LNA became unmatched, leading to a Low-IF output amplitude of −50 dBm (1 mV in a 50 ⍀ load), as shown in Fig. 7.11 (instead of the expected −30 dBm). The quadrature errors, obtained by using the procedure described above (in 7.2.2) are of the order of 1◦ (Fig. 7.11) in 4 measured die samples. A plot of the measured phase-noise is shown in Fig. 7.12. Since this noise has been measured after frequency division by two, 6 dB must be added to find the actual phase-noise of the oscillator-mixer (assuming a noiseless frequency divider). Phase-noise close to the carrier is suppressed by the PLL, so it is the PLL reference

130

7 Measurement Results 0 –10

Amplitude [dBm]

–20 –30 –40 –50 –60 –70 –80 2.45

2.46

2.47

2.48

2.49 2.5 2.51 Frequency [GHz]

Fig. 7.10 Divider by two output at 2.5 GHz

Fig. 7.11 Low-IF (10 MHz) quadrature outputs

2.52

2.53

2.54

2.55

7.4 Quadrature Oscillator-Mixer

131

PLL reference oscillator

quadrature oscillator

Fig. 7.12 Oscillator phase-noise

oscillator that defines the output phase-noise within the PLL bandwidth, which is about 100 kHz. Above 1 MHz, a slope of −20 dB/decade is observed, which is due to the quadrature oscillator, and this is the phase-noise that we wish to measure. The phase-noise of the oscillator-mixer circuit (functioning as an oscillator only) at 10 MHz offset from the carrier (at 5 GHz) is about −114 dBc/Hz. The phase-noise was measured with an HP phase-noise measurement system, which has a better accuracy than a spectrum analyzer (the equipment used has an accuracy of ±2 dB). Using this equipment we do not need to calculate the carrier power and to take into account the spectrum analyzer bandwidth. For a comparative study between this oscillator and other oscillators presented in the literature, we use the following FOM (figure-of-merit) [107]:  FOM = Lmeasured + 10 log

⌬f f

2

PDC Pr e f

 (7.7)

where PDC is the DC power dissipated by the oscillator; Pr e f is a reference power level (typically 1 mW). This FOM contains information on phase-noise, oscillator frequency, offset frequency, and power dissipation. The results in Table 7.5 show that the proposed quadrature oscillator performance is comparable to that of other state-of-the-art RC oscillators. This test-circuit used without modulation is used to make a comparison with other state-of-the-art RC oscillators (either single or cross-coupled). The reason to use this circuit (and not those in Sections 7.2 or 7.5) is that we had access to a more precise measurement system for this circuit (at TU Delft).

132

7 Measurement Results Table 7.5 Comparison of state-of-the-art RC oscillators

Ref.

Lmin [␮m]

Vcc [V]

f [GHz]

⌬f [MHz]

L(⌬ f ) FOM f max [dBc/Hz] [dBc/Hz] [GHz]

[108] [109] [36] [36] [110] [37] [37] [111] This work

1.2 0.6 0.5 0.5 0.35 0.25 0.25 0.18 0.25

5.0 3.0 3.0 3.0 3.3 2.5 2.5 1.8 2.5

0.93 0.90 2.20 0.92 0.97 1.33 5.43 3.52 5

0.1 0.6 5.0 5.0 1.0 1.0 1.0 4.0 10

−83 −117 −109 −102 −117 −112 −99 −106 −114

−154 −165 −158 −151 −158 −164 −154 −153 −154

0.93 1.20 2.20 0.92 0.97 1.33 5.43 3.52 5.3

f max f min 2.9 1.6 – – – – 1.3 35 1.23

7.5 Comparison of Quadrature LC and RC Oscillators In order to verify the conclusions concerning the phase-noise of RC oscillators and, in addition, to compare its noise with that of LC oscillators, we designed an RC and an LC oscillator for the same frequency (5 GHz), using the same technology (TSMC CMOS 0.18 ␮m), and we had them fabricated on the same chip. The quadrature RC oscillator, with the schematic of Fig. 7.1, has R = 100 ⍀, (W/L) = 75 ␮m/0.18 ␮m for the M transistors, (W/L) = 100 ␮m/0.18 ␮m for the M SL transistors, C = 300 fF, I = 3 mA, and I SL = 500 ␮A (weak coupling) or I SL = 3 mA (strong coupling). The supply voltage is 3 V. The quadrature LC oscillator has the circuit schematic of Fig. 7.4, and uses integrated inductors. It has (W/L) = 75 ␮m/0.18 ␮m for the M transistors, (W/L) = 75 ␮m/0.18 ␮m for the M SL transistors, L = 2 nH, Q 0 = 5, I = 1 mA, and

LC Oscillator

Fig. 7.13 Die photo of the test circuit with the two oscillators

RC Oscillator

7.5 Comparison of Quadrature LC and RC Oscillators

133

Table 7.6 Phase-noise @ 1 MHz of the RC and LC oscillators Coupling

RC

LC

Sim.

Meas.

Sim.

Meas.

Weak I S L = 0.5 mA Strong I S L = 3 mA

−90 dBc/Hz −98 dBc/Hz

−87 dBc/Hz −97 dBc/Hz

−101 dBc/Hz −92 dBc/Hz

−101 dBc/Hz −96 dBc/Hz

I SL = 500 ␮A (weak coupling) or I SL = 3 mA (strong coupling). The supply voltage is 1.8 V. The die photo of the test circuit with the two oscillators is shown in Fig. 7.13. The total die area is 1 mm × 2 mm; this large area is necessary to allow the use of wafer probes. The die area without pads and metal fills is 0.012 mm2 for the RC oscillator and 0.0945 mm2 for the LC oscillator. The area of the LC oscillator is 7.7 times higher than that of the RC oscillator. Strong coupling ensures accurate quadrature relationship for both oscillators. Yet, for the RC oscillator, strong coupling improves both the quadrature relationship and the phase-noise performance (Tables 7.6 and 7.7). In the LC oscillator, strong coupling reduces the quadrature error, but increases the oscillator phase-noise. This has been predicted in [20, 63] and is confirmed experimentally here. The outputs of the oscillators were measured using G-S-G probes. Losses in the coaxial cables and DC blocks connecting the on-wafer probes to the scope were around 7 dB at 5 GHz. The measured output amplitude of each oscillator was close to 100 mV. The quadrature accuracy (Table 7.7) was measured using the starting edge detection capability of the HP54120B digitizing oscilloscope. Figures 7.14 and 7.15 show the output waveforms of the two quadrature oscillators. We can compare these oscillators using the conventional figure-of-merit [107] (equation (7.7)). We also use a recently proposed figure of merit FOMA [112] that includes the die area:    Δ f 2 PDC Ar e f (7.8) FOMA = Lmeasured + 10 log f Pr e f Achi p where Achi p is the circuit area in mm2 , and Ar e f is a reference area (1 mm2 ). Table 7.8 shows that, with weak coupling, the LC oscillator is better than the RC oscillator in terms of both FOM and FOMA. Yet, if one uses strong coupling to reduce the quadrature error, the noise performance of the RC oscillator improves Table 7.7 Quadrature error of RC and LC oscillators Coupling

RC

LC

Weak I S L = 0.5 mA Strong I S L = 3 mA

2.7◦ 1◦

3.5◦ 1.5◦

134

Fig. 7.14 RC-oscillator quadrature outputs

Fig. 7.15 RC-oscillator quadrature outputs

7 Measurement Results

7.6 Conclusions

135 Table 7.8 FOM and FOMA for the RC and LC oscillators

Coupling Weak I S L = 0.5 mA Strong I S L = 3 mA

RC

LC

FOM

FOMA

FOM

FOMA

−145 dBc/Hz −154 dBc/Hz

−164 dBc/Hz −173 dBc/Hz

−168 dBc/Hz −159 dBc/Hz

−178 dBc/Hz −169 dBc/Hz

significantly, and that of the LC- oscillator is degraded. The FOM value improves, but it remains worse for the RC oscillator. The FOMA value, however, is better for the RC oscillator with strong coupling. It should be noted that the noise value in the LC oscillator is strongly dependent on the inductors’ Q and circuit topology. Better values may be obtained for LC oscillators different from those used here. Since in our example the simplest circuits have been chosen for both the LC and RC oscillators, the comparison presented here is believed to be reasonably fair.

7.6 Conclusions In this chapter we confirm by measurements on different quadrature relaxation oscillators that the coupling block is critical concerning quadrature error and phase-noise. The first prototype is an RC relaxation oscillator. In these oscillators the coupling gain should be high to force oscillators to be in precise quadrature (mismatches and other disturbances will have then only a second order effect). Moreover, by increasing the coupling gain, the oscillator will improve the phase-noise performance. However, we cannot increase this gain indefinitely, because this will increase the power dissipation and cause some reduction in the oscillation frequency. In a second prototype we show that in quadrature LC oscillators there is a trade off: the coupling gain should be high to force a precise quadrature, but increasing the coupling gain will increase the phase-noise. The third prototype is used to demonstrate the oscillator-mixer concept, using an RC relaxation oscillator. The measurements confirm that this class of oscillators can perform wideband mixing. This oscillator-mixer circuit is well suited to be used as part of a double-conversion receiver with a low intermediate frequency, where very accurate quadrature signals are necessary to reject the image signal. Measurements show an error of the order of 1◦ for four different die samples of the oscillator-mixer. The phase-noise measured is comparable to that of state-of-the-art RC oscillators. The last two prototypes, at 5 GHz, show that in RC oscillators the coupling reduces both the phase-noise and quadrature error, whereas in LC oscillators the coupling reduces the quadrature error, but increases the phase-noise. Figure of merit values indicate that quadrature RC oscillators may be a viable alternative to LC oscillators when area and cost are to be minimized.

Chapter 8

Conclusions and Future Research

Contents 8.1 8.2

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

8.1 Conclusions In recent years, wireless communications have been developed due to the huge demand for mobile equipments. Associated with the mobility, equipment low size and cost are important requirements. Therefore, avoiding the use of discrete components in modern transceivers is an important topic of research. Full integration of modern transceivers requires very accurate quadrature signals. There are two basic receiver front-end architectures: heterodyne, which uses one or more intermediate frequencies, and homodyne, without intermediate frequency. The main drawback of heterodyne receivers is that the image frequency band must be removed with external filters, which does not allow full integration. The homodyne approach allows full integration, since it does not have the image problem, but the direct conversion to the baseband imposes severe restrictions (1/f noise, DC offsets, intermodulation distortion, etc.), which are not present or can be easily avoided in the heterodyne approach. Since both conventional approaches have advantages and drawbacks, why not use an approach that uses the best features the two? This is what does the low-IF receiver. This is an heterodyne receiver, but it does not use filters to remove the image frequency band: an image reject technique is used instead that allows the use of a very low intermediate frequency. Since external filters are not used, full integration of the receiver on a single chip is possible; since there is no conversion to the base-band, the problems of homodyne receivers are avoided. The main issue of this approach is that it requires very accurate quadrature outputs from the local oscillator to cancel the image. The amount of image rejection has a direct relationship with the quadrature error. This justifies the development of new techniques to provide very accurate quadrature outputs. L.B. Oliveira et al., Analysis and Design of Quadrature Oscillators,  C Springer Science+Business Media B.V. 2008

137

138

8 Conclusions and Future Research

In this book we study and compare three types of quadrature oscillators: crosscoupled RC and LC oscillators, and the two-integrator oscillator (that has inherently quadrature outputs). These three types of oscillators, despite being conceptually different, have one common feature: they are closed-loop structures (conventional architectures are open-loop), which leads to accurate quadrature. RC oscillators are known for their poor phase-noise performance when compared with LC oscillators, reason why they have been neglected. Although this is true for a single oscillator, it is not for cross-coupled oscillators. In this book we study in detail the quadrature relaxation oscillators in terms of their key parameters: oscillation frequency, amplitude, quadrature relationship, and phase-noise. The effect of mismatches and other disturbances is attenuated by the soft limiter gain, becoming a second order effect, and allowing a very accurate quadrature. We derive equations for the oscillator amplitude, which show that in the presence o mismatches the amplitudes change to preserve the quadrature accuracy. We point out that the coupling reduces the phase-noise. In relaxation oscillators both the quadrature error and the phase-noise are reduced due to the coupling. The influence of coupling in the oscillator performance was confirmed by measurements on a 2.4 GHz quadrature relaxation oscillator. Single LC oscillators are widely used in applications in which very low phasenoise is necessary. They are now the first choice in the design of modern transceivers. In this book we consider the coupling of these oscillators to obtain quadrature outputs. We calculate the oscillation frequency of two coupled oscillators, and show that a frequency change due to the coupling affects the Q-factor, and, therefore, degrades the phase-noise. Comparing RC and LC quadrature oscillators, we conclude that coupling in relaxation oscillators improves simultaneously the quadrature accuracy and phasenoise, whereas in LC oscillators the coupling improves the quadrature, but increases the phase-noise. Thus, in a quadrature LC oscillator there is a compromise in choosing the strength of coupling, which does not exist in the quadrature RC oscillator. Thus, quadrature RC oscillators can be an alternative to quadrature LC oscillators. The degradation of phase-noise due to coupling was confirmed by measurements on a 1 GHz quadrature LC oscillator. The advantages of quadrature relaxation oscillators are left out of the conventional FOM (figure of merit), which considers only phase-noise and power consumption. There, is however, another figure of merit, FOMA, that takes also the area into account. We have used both FOM and FOMA to compare experimentally an RC and an LC cross-coupled oscillator at 5 GHz. RC oscillators have lower area, since they do not require inductors that need a large silicon area, and this advantage is evidenced by the FOMA values. RC oscillators can be built with a low cost technology (they do not need special RF options, several metal layers, and thick top metal layer, in order to improve the inductor quality factor). The conclusion of our experimental comparison is that quadrature RC oscillators can be a suitable alternative to RC oscillators when area and cost are to be minimized.

8.2 Future Research

139

The tuning range of the previous cross-coupled oscillators (RC and LC) is typically lower than 20%, which is an important limitation, since nowadays applications with wide frequency band are being developed. The two-integrator oscillator is a single oscillator with quadrature outputs (it is not formed by two coupled oscillators). Its phase-noise is comparable to that of a relaxation oscillator. The main advantage of this oscillator is the wide tuning range (close to one decade), which makes it a good solution for wideband or multi-standard applications. In this book we show that the mixing function can be incorporated in the three types of quadrature oscillators, and that the quadrature relationship is preserved. This approach avoids separate mixers, whose mismatches are responsible for a first order error in the quadrature relationship, and has the advantage of saving area and reducing the power consumption of the complete transceiver. We demonstrate the oscillator-mixer concept with measurements on a 5 GHz quadrature relaxation oscillator-mixer prototype.

8.2 Future Research We suggest the following further research topics in the area covered by this book:

r

r

r

r

Quadrature oscillators are key blocks in the design of transmitters and receivers. A natural continuation of the work in this book is to include the oscillators in complete receivers (especially low-IF or direct conversion receivers) and evaluate the improvement in system performance. The oscillators should also be included in transmitters to compare their advantages with the conventional approaches. In this book we show that the usual coupling of LC oscillators, using the first harmonic of the signal, degrades the oscillator phase-noise. An important topic of research is the development of new coupling methods (such as second harmonic coupling) to eliminate this drawback. This work was already started, and has led to some publications [23, 24]. The performance of coupled LC oscillators is non-linear, but, traditionally, linear models are used to analyse the oscillators. A new model is required in which the oscillator is treated as a non-linear circuit. A possible approach is to consider the old non-linear differential equations of the van der Pol oscillator. The study of this approach has already been started [23–25]. A complete study of the stability of oscillations is required to find which of the different mathematically possible oscillation frequencies are applicable. The main purpose of this book was the study of quadrature oscillators; however, we also started the study of the oscillator-mixer concept and we presented possible implementations. This study was left at an initial stage. A comprehensive study of the mixing, at circuit level, should be done in terms of key parameters (for example, linearity, and noise figure); a comparative study with conventional mixers to evaluate the advantages and disadvantages is also required.

140

r r

8 Conclusions and Future Research

In Chapter 6, the study of the two-integrator oscillator was confirmed by simulation only. A validation by measurements on a test chip of the wideband properties of this oscillator should be done. In recent years the reduction of power consumption and of supply voltage (required by modern sub-micron technologies) is an important area of research. An extension of the study in this book considering modern technologies, with low voltage supply and with low power consumption is an important topic of future research.

Appendix A

Test-Circuits and Measurement Setup

Contents A.1 A.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quadrature RC and LC Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.1 Test Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.2 Measurement Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Quadrature Relaxation Oscillator-Mixer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.1 Test Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.2 Measurement Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

141 141 141 142 144 144 151

A.1 Introduction In this appendix we describe in a more detailed way the test-circuits and the measurement setup for the five designed prototypes. For the quadrature RC and LC oscillators we measure the quadrature error and phase-noise as a function of the coupling gain. The test-circuits are similar and the measurement setup is the same. This is used to test four of the five prototypes mentioned in Chapter 7: a quadrature RC oscillator at 2.4 GHz, a quadrature LC oscillator at 1 GHz and two quadrature RC and LC oscillators at 5 GHz. The remaining prototype is the oscillator-mixer at 5 GHz. We include it in a PLL and combine this with an LNA and low-pass filters, for the validation of the oscillator-mixer concept, as explained below.

A.2 Quadrature RC and LC Oscillators A.2.1 Test Circuit In order to have more control of the circuit variables we have an external biasing circuit. We are able to control the tail current of the coupling differential-pairs (to change the coupling gain), the oscillator bias current (to tune the oscillator), and the buffers bias current (to control the output matching). For this purpose we use, respectively, the three input terminals, “coupling”, “VCO”, and “buffer”, marked in Fig. A.1. 141

142

Appendix A

Fig. A.1 Test-circuit of the cross-coupled RC and LC oscillators

Buffer

V +outI

Buffer

V −outI

Buffer

V +outQ

Buffer

V −outQ

I

Quadrature Oscilltor Q Coupling

VCO

VCC

Vin1

Vin2

M2

Vin3

M2

M2

Vin4

M2

IB

Vout1

M1

M1

Vout2

M1

Vout3

M1

Vout4

M1

Fig. A.2 Output buffers

With the test-circuit represented in Fig. A.1 we are able to measure the phasenoise and quadrature error as a function of the coupling current. The buffers at the oscillator output are needed to drive the 50 ⍀ loads. The buffers are common-drain (source-follower) stages, represented in Fig. A.2. The buffers are followed by 50 ⍀ microstrip lines on the test printed circuit board (PCB), and external DC blocking capacitors are used. When the buffers are biased with I B = 2 mA, the oscillator output voltage is about 150 mV. The buffer implementation has M1 transistors with (W/L) = 20 ␮m × 1 ␮m and M2 transistors with (W/L) = 30 ␮m × 0.35 ␮m.

A.2.2 Measurement Setup The complete measurement setup for measuring the oscillator quadrature error is shown in Fig. A.3. The three current sources to bias the oscillators, coupling blocks, and buffers can be adjusted by using external resistors.

Appendix A

143 6V

Batteries

Voltage Regulator

3V

Digital oscilloscope VCC

CHIP

VCO Coup

Rvar

GND

Rvar

Buffer

I+ I–

Trigger

Q+

CH2

Q

CH1

–

Rvar

50 Ω

VCC

Fig. A.3 Setup for measurement of the quadrature error

The oscillator has four outputs, but the oscilloscope has only two inputs. A problem is to trigger the oscilloscope, because we use a sampling oscilloscope. These instruments do not have the ability to synchronize using the input signals. Clock recovery modules are needed if we do not have access to a trigger source. Since in our circuit we have 4 outputs, we measure two outputs with the oscilloscope, use another output as trigger, and connect the last output to a 50 ⍀ load. It was found that it is very important to have the four outputs connected to 50 ⍀ loads to have accurate quadrature. To prevent any interference from entering the system through the power supply, a battery supply was used. We use 4 batteries of 1.5 V and the resulting 6 V voltage is applied to a voltage regulator to produce the desired 3 V supply. To connect the oscillator outputs to the oscilloscope we use 50 ⍀ microstrip lines on the board, SMA connectors, and 50 ⍀ cables between the board and the oscilloscope. A photo of the measurement board and batteries is shown in Fig. A.4. The circuit uses only a few external components for biasing and DC blocking. We have used four 1 nF SMD capacitors for the four buffers DC blocking. For the power supply decoupling we use 2 capacitors: a high frequency bypass capacitor of 100 pF and a low frequency bypass capacitor of 0.47 ␮F. The following measurement equipment is used:

r r

Oscilloscope – Agilent 83484A communications analyser (50 GHz bandwidth). This is used to observe the two oscillator outputs in quadrature. Spectrum analyzer – Tektronix real time spectrum analyzer RSA2208A (8 GHz bandwidth). This is used to observe the output spectrum of the oscillator and measure the phase-noise.

144

Appendix A

Fig. A.4 Measurement setup

The organization of the test circuit (Fig. A.1) and the measurement setup (Fig. A.3) are the same for the quadrature RC and LC oscillators. The only difference is that the 1 GHz quadrature LC oscillator uses external inductors. The measurements results obtained with this setup have been presented in Chapter 7.

A.3 Quadrature Relaxation Oscillator-Mixer A.3.1 Test Circuit The PLL, LNA, and buffer are designed to support the testing of the oscillatormixer. These circuits are well known and their design does not present major problems. The proposed quadrature oscillator-mixer is used in downconversion as shown in Fig. A.5. An RF signal at 5.01 GHz is converted to a low IF (10 MHz). After the oscillator-mixer, the I and Q outputs are filtered (for channel selection) and buffered, so that off chip loads can be driven. To drive the oscillator-mixer, off-chip signals have to be applied to the chip. This can be done by using a low-noise amplifier (LNA) with 50 ⍀ input impedance and two current outputs. These two currents are the bias currents of the two soft-limiters. Relaxation oscillators are known to be noisy. To reduce the phase-noise, and to prevent frequency pulling by the input signal, the oscillator-mixer is inserted in a PLL. To keep the test circuit as simple as possible, the phase-detector and the loop filter will be realized externally. To prevent cross-talk from a strong in-band signal to the input of the LNA, the oscillator output signal has its frequency divided by two inside the chip.

Appendix A

145 Dummy Buffer

RF Input LNA

I

Buffer

LPF

Q

Buffer

LPF

IF-I

Oscillator Mixer IF-Q VCO

Bias Circuit

Buffer

Divider by 2

Buffer

Phase Detector Test-Chip

Loop Filter

Reference Generator

Fig. A.5 Test environment for the oscillator-mixer

Since the oscillator-mixer is easily tuned, it is also sensitive to noise. For this reason, an on-chip bias block is used, which generates bias currents for all the circuits on the chip. In this section we present the circuit schematics of the blocks in Fig. A.5 (except the oscillator-mixer). All the circuits operate from a 2.5 V supply voltage.

A.3.1.1 LNA The purpose of the LNA is to convert an input signal into the tail currents of the limiters in the oscillator/mixer circuit. The LNA is as simple as possible, since the main purpose of the design is to show the possibilities of the oscillator-mixer. The LO frequency is different from the RF frequency. Receiver desensitization and DC offset due to LO self-reception are not a problem here, as they would be in direct conversion front-ends [113]. Therefore, it is not necessary to use a fully differential LNA. A single-ended implementation has lower consumption and a lower noise figure [1]. An off-chip single-ended to differential conversion by means of a balun is not required, since the LNA has a single-ended input.

146

Appendix A

VCC

IBIAS

Qa Osc SL1

Osc SL 2

R2

R1 Vin

LS

Cin

Q1

Q2

LE R

Q3

LE

R3

R CE

Cfilt

CE

Fig. A.6 LNA

The LNA schematic is shown in Fig. A.6 [1]. It consists of two inductively degenerated common emitter (CE) stages (Q transistors with emitter area 0.32 ␮m × 10 ␮m). Connected to the bases, is an inductor L s (1.8 nH), which lowers the frequency at which the LNA input impedance is real. It is implemented using a piece of microstrip on the printed circuit board on which the chip is mounted. This type of LNA has reasonable noise and intermodulation performance, and offers a simple means to achieve impedance matching at the input. The input impedance is 50 ⍀ for easy interface with the measuring equipment. The frequency at which the input impedance is real, is approximately 1 $ L + L s C␲ 2 e

␻20 = # 1

(A.1)

where C␲ is the small-signal base-emitter capacitance of the transistors. One problem with this type of LNA is its biasing. Reasonably matched transistors, inductors, and collector currents are required. Accurate collector current matching is achieved here by means of resistive emitter degeneration at DC. The degeneration is bypassed at the frequencies of interest by using a 10 pF capacitor C E in series with L E . The bias current is obtained by mirroring a lower current. Transistor Q 3 with emitter area 0.32 ␮m × 2 ␮m is the input of the current mirror, and Q a (emitter area 0.32 ␮m × 2 ␮m) reduces the inaccuracy due to the base

Appendix A

147

currents. All resistors that need to be matched are composed of series and parallel connections of unit-size resistors. The reason for the presence of the resistors R1 and R2 and the choice of their values will now be explained. With I B I AS = 400 ␮A, we wish Q 1 and Q 2 with 5 × 400 ␮A = 2 mA. This can be accomplished by choosing Q 1 and Q 2 with an emitter area 5 times larger than that of Q 3 . To improve the matching between the emitter currents, some resistive emitter degeneration has been included, and the voltage over the emitter resistors, R = 50 ⍀ and R3 = 250 ⍀) is 0.1 V. To avoid inaccuracy due to the base currents, Q a is added. The impedance that is seen at the base of the transistors of the current mirror is very low now, due to the local feedback around Q a and Q 3 . To increase it to an acceptable value, R1 = 1 k⍀ is included. This value is much higher than 50 ⍀, the intended input impedance. However, the voltage drop over R1 , due to the base currents of Q 1 and Q 2 , would lead to a mismatch of the collector currents of Q 1 and Q 2 with respect to that of Q 3 . Assuming a current gain of 100, the base current is 20 ␮A and the voltage drop over R1 is 40 mV. To compensate this, a resistor R2 = 10 k⍀ is inserted in series with the base of Q 3 , which has the same voltage drop (namely 400 ␮A/100 × 10 k⍀) as that over R1 . We also add a capacitor C f ilt (10 pF) to reduce the noise of the bias network. Simulation of the LNA has been performed to obtain its noise, distortion, gain, and input impedance. ESD input protection diodes and pad models (C4 ball) have been incorporated in the simulation. The influence of the DC decoupling capacitor (Cin ) at the bases of the transistors turned out to decrease the input impedance of the LNA. This was solved by increasing the value of the inductances in the emitter to a final value of L E = 0.72 nH (Fig. A.6). The noise figure is about 2.5 dB, and the transcondutance is about 25 mA/V. The simulated IIP3 is +7 dBm.

A.3.1.2 Frequency Divider The frequency divider divides the oscillator frequency by 2, so that it is in a different frequency band than the front-end input signal, thus avoiding interferences. The frequency divider is implemented with two current-mode logic D-type flip-flops. A detailed explanation of the circuit performance can be found in [1]. The bias currents have been chosen such that the total divider current is low with respect to that of the oscillator-mixer, which minimizes the overhead power consumption. The voltage swing in the resistors is about 300 mVpp, to ensure reliable switching of the differential pairs in the latches. The schematic of the frequency divider is shown in Fig. A.7 [1]. The emitter area of the transistors is 0.32 ␮m × 2.5 ␮m for Q and Q 1 , 0.32 ␮m × 10 ␮m for Q 2 , 0.32 ␮m × 20 ␮m for Q 3 , and 0.32 ␮m × 5 ␮m for Q 4 , R1 = 500 ⍀, R2 = 125 ⍀, R3 = 62.5 ⍀, RC = 250 ⍀, and C = 2 pF. The total bias current is 3.4 mA, and simulations showed reliable operation up to 10 GHz with an amplitude of 150 mV.

148

Appendix A VCC

RC

RC

RC

RC Q4

Q

Q

Q

Q

Q

− V in1

+ V in1

Q

Q

Q

Q

+ V in1

Q4 Vout

− V in1

Q

Q

Q

IBIAS

Q1

Q2 R1

C

Q3

Q2 R2

R2

Q3

R3

R3

Fig. A.7 Frequency divider

A.3.1.3 Bias Current Generator The bias circuit generates currents I B I AS for the circuits in Figs. A.6–A.10. These currents are independent from of the supply voltage, to reduce noise coupling through the power supply. There are no connections off-chip to sensitive parts of the bias current generator, which further reduces sensitivity to noise. The bias current generator schematic is shown in Fig. A.8 [114]. VCC R1

R1

R1

C M

M

Rstart

mI

I

Q

m sections in parallel

Q Qref 1 Q

Fig. A.8 Bias current generator

M

Qref 2 R

Appendix A

149 VCC

Fig. A.9 Divider buffer

Vout1 V +in1

Q

Q

– V in1

I BIAS

Q1

Q

C

R1

R

It can be shown that current I is [114] I =

VT ln(n) R

(A.2)

where VT = kT /q, k is Boltzmann’s constant, T is the temperature in Kelvin, q is the electron charge, and n is the emitter-area ratio of the transistors Q r e f 2 and Q r e f 1 . In this design, Q r e f 1 has an emitter area of 0.8 ␮m × 10 ␮m, and Q r e f 2 is composed of 8 of these transistors in parallel. With the parameters chosen, I = 100 ␮A at 20◦ C. Current I is PTAT (proportional-to-absolute-temperature) if resistor R is temperature-independent. Here, a resistor has been used which has a positive temperature coefficient, so the current is approximately constant: over the temperature range 0–70◦ C the deviation from the nominal value is about ±7%. VCC

Vin 1

Vin 2

Q

Vin 3

Vout 1

IBIAS

Q

Q1 R1

Q

C

Fig. A.10 Oscillator-mixer buffer

R

Vin 4

Q

Vout 2

Q

R

Q

Vout 3

Q R

Vout 4

Q R

150

Appendix A

To keep the dependence of the current on process variations as small as possible, R is a wide resistor (6 ␮m). The 3-sigma variation of the resistance value is 10%. In combination with variation due to temperature, the reference current can deviate up to 15% from the nominal value, which was found acceptable. A start-up circuit, composed of Q 1 –Q 3 and Rstart with a value of 100 k⍀ has been included. The start-up current is a few nA, which is small enough not to compromise accuracy. To reduce the influence of shot noise and to improve matching, source degeneration resistors are used in the current mirrors. Multiples m of the minimum current are generated by putting m unit current sources in parallel, as shown in Fig. A.8. The noise at high frequencies is reduced by a capacitor of 30 pF (Fig. A.8). To achieve a high current source output impedance, long transistors (2 ␮m) were used. Each PMOS is composed of 5 sections, 10 ␮m wide. The values of resistors are R = 536 ⍀ and R1 = 1 k⍀. A.3.1.4 Buffers The frequency divider output buffer is represented in Fig. A.9. This buffer converts the divider differential output into a single-ended output: the output current at terminal Vout1 is converted to voltage with an external resistor of 50 ⍀ connected to VCC ; Vout1 is connected to the phase-detector. The currents were dimensioned according to the load and the peak-to-peak output voltage of the divider, which is about 300 mV. The implementation uses Q and Q 1 transistors with emitter area 0.32 ␮m × 16.8 ␮m and 0.32 ␮m × 4.2 ␮m, respectively, R = 50 ⍀, R1 = 200 ⍀, and C = 2 pF. The oscillator-mixer output buffers are common-collector stages. The set of four buffers has the circuit in Fig. A.10. Transistors Q and Q 1 emitter areas are 0.32 ␮m× 8.6 ␮m and 0.32 ␮m × 4.3 ␮m, respectively, R = 50 ⍀, R1 = 100 ⍀, and C = 2 pF. These buffers are followed by low-pass filters. The low-pass filters (Fig. A.11) have 5th order Butterworth response with cutoff frequency 1 GHz for 3 dB attenuation. The component values are L 1 = 12.221 nH, L 2 = 11.04 nH, L 3 = 2.472 nH, C1 = 5.39 pF, and C2 = 2.85 pF. For good matching these filters are on-chip. In a practical implementation these filters can be substituted by much smaller active filters, but in this design the main concern is to measure the quadrature error, so we have a passive low-noise and linear filter. L1

L3

L1

Vin

Vout C1

Fig. A.11 Low-pass filter

C2

Appendix A

151

A.3.1.5 External blocks The remaining external blocks to build the Phase-lock-loop (PLL) are the phasedetector and the loop filter. These blocks are connected externally. This decision was made to concentrate efforts on the design of the oscillator-mixer, due to time limitations. The phase-detector is a commercial mixer (Miteq DMX0418L), which can be used up to 18 GHz. This is followed by a first-order filter with an input impedance of 50 ⍀, that has one pole and one zero. Since the oscillator frequency is controlled by a current, the filter is followed by a simple transconductance amplifier with a transconductance of about 1 mA/V. A signal-independent current, for tuning the oscillator to its nominal value, can be set by means of a potentiometer. A.3.1.6 Layout The layout had to satisfy an important constraint imposed by the technology: the C4 bondpads were not available in the fabrication run used, and had to be replaced by wirebond pads, resulting in additional inductances and in changes in the layout.

A.3.2 Measurement Setup The measurement setup is shown in Fig. A.12. The transformers convert the balanced output signals from the chip to single-ended signals, suitable for measurement by an oscilloscope. During first measurements, it was found that the output signals were disturbed by electromagnetic interference. Therefore it was decided to shield the chip and the PLL loop filter. These were placed inside an aluminum box, and signals were taken in and out of the box by BNC and SMA connectors (Fig. A.13). The power-on

RF input

Inside Metal BOX

2.5V VCC LNA IN DIV OUT

CHIP VCO GND IN

Digital oscilloscope

I+ I–

CH1

Q+

CH2

Q–

50W

PLL Reference oscillator

5V Loop filter Phase Detector

5V 2.5V

Fig. A.12 Setup for measurement of quadrature error

Battery supply

152

Appendix A

Fig. A.13 Aluminum box containing the oscillator-mixer

switch and a variable resistor for tuning the oscillation frequency are placed outside the main metal box, but they are shielded inside a tin plate box with a lid. To avoid any interference from entering the system through the power supply, a battery supply was used. The circuit uses only a few external components for biasing and for DC blocking of the oscillator outputs. We have used a DC blocking 800 pF SMD capacitor. Single-ended inputs and outputs have been used for RF signals, to avoid differential to single-ended conversion on the board (for connection to measuring equipment). Only the low-IF is led off the chip differentially, since differential active measurement probes are available (these are modeled, for simulation, by a 50 ⍀ resistance in parallel with a 7 pF capacitance). The supply voltage is bypassed using a 0.47 ␮F capacitor (low frequencies) in parallel with a capacitor of 100 pF (high frequencies). Due to the unavailability of the C4 pads for flip-chip bonding, we used wire bonding to the PCB. The RF signals are led to and from the chip using 50 ⍀ microstrip lines and SMA connectors. The low-IF outputs have lower frequency and are connected to transformers for conversion from differential to single-ended. The following measuring equipment is used: Signal Generator – HP ESG – D4000A (250 kHz–4 GHz). This is used to generate the PLL reference signal applied directly to the mixer.

Appendix A

153

Signal Generator – HP 8665B (0.1 MHz–6 GHz) This is used to generate the RF signal. Amplifier – HP 8437A (100 kHz–3 GHz) This is used to amplify and isolate the signal of the divider by 2. Oscilloscope – Fluke PM 3380A. This is used to observe the PLL output and check if the PLL is locked. Oscilloscope – Agilent Mixed Signal Oscilloscope 5462 2D (100 MHz–200 Ms/s) This is used to observe the two IF outputs. Spectrum analyzer – HP 8566B (100 Hz–2.5 GHz or 2 GHz–22 GHz) This is used to observe the divider by two output. Spectrum analyzer – HP 8568B (100 Hz–1.5 GHz) This is used to observe the IF output. The measurements results for this oscillator-mixer have been presented in Chapter 7.

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Index

A Aluminum box, 152 Amplitude modulation, 77 Amplitude noise, 19, 84 B Barkhausen criterion, 17 BiCMOS technology, 129 C CMOS tecnhology, 121, 132 D Duty-cycle, 72 E Excess noise factor, 22

J Jitter, 19 L

L (phase-noise), 19 LC oscillator, see Oscillator, LC LC tank, 83 Leeson’s formula, 19 LNA, 145 Low-IF receiver, 2, 11 Figure of merit FOM, 131, 133 FOMA, 133 M Mixers, 26 Modulation, see Amplitude modulation

F Frequency division, 33–34

N

G GSM, 93

Noise factor, 27 jitter, 19 phase-noise, 18 transfer function, 107

H Hartley architecture, 12 Havens’ technique, 34–35 Heterodyne receiver, 1, 8 Homodyne receiver, 2, 10 I Image rejection, 9, 13, 122 Inductor bondwire, 125 external, 125 integrated, 93, 132 RF options, 93 Intermodulation, 27

O Oscillator LC, 25 figure-of-merit, 131 quasi-linear oscillator, 24 RC (relaxation), 24, 39 strongly non-linear, 24 two-integrator, 105 Oscillator-mixer LC, 96 RC, 95 two-integrator, 113

161

162 P Phase difference, 73 Phase-error quadrature LC oscillator, 91 quadrature RC oscillator, 138 two-integrator oscillator, 111 Phase-noise 1/ f noise, 11 definition, 18 quadrature LC oscillator, 92–96 quadrature RC oscillator, 60 single LC oscillator, 85 single RC oscillator, 60 two-integrator oscillator, 107–111 PLL, 144 bias current, 148 filters, 150 frequency divider, 147 LNA, 145–146 phase-detector, 151 phase-noise, 131 Q Quadrature accuracy, see Phase-error Quadrature LC oscillator implementation, 85 measurement setup, 119 phase-error, 91 phase-noise, 95 Q degradation, 94 test-circuit, 131 Quadrature oscillator-mixer high level, 64 implementation, 75 measurement setup, 151 oscillation frequency, 72 phase-error, 74 phase-noise, 79 test-circuit, 131

Index Quadrature RC oscillator high level, 41–42 implementation, 44 measurement setup, 142–144 phase-error, 41, 46 phase-noise, 60–61 test-circuit, 131, 142 Quality factor, Q coupled LC oscillators, 93 definition, 19 two-integrator, 109 R RC-CR network, 31 RC relaxation oscillator high level, 38 implementation, 39 Resonance frequency, 25, 84 S Single LC oscillator, 82 Single RC oscillator, 56 T Transmitter, 15 Two integrator oscillator implementation, 104–107 non-linear, 100 oscillator-mixer, 113 phase-error, 111 phase-noise, 110, 112 quasi-linear, 102 tuning range, 113 W Weaver architecture, 14 White noise, 22 Z Zero-IF receiver, see Homodyne receiver