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 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