Analysis and design of quadrature oscillators

ANALYSIS AND DESIGN OF QUADRATURE OSCILLATORS

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Analysis and Design of QuadratureOscillators

by

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

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

Igor M. FilanovskyUniversity of Alberta, Canada

Chris J.M. VerhoevenTechnical University of Delft, The Netherlands

and

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

123

Dr. Luis B. OliveiraINESC-IDRua Alves Redol 91000-029 [email protected]

Dr. Jorge R. FernandesINESC-IDRua Alves Redol 91000-029 [email protected]

Dr. Igor M. FilanovskyUniversity of AlbertaDept. Electrical &Computer Engineering87 Avenue & 114 StreetEdmonton AB T6G 2V42nd Floor [email protected]

Dr. Chris J.M. VerhoevenDelft University of TechnologyMekelweg 42628 CD [email protected]

Dr. Manuel M. SilvaINESC-IDRua Alves Redol 91000-029 [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 transmittedin any form or by any means, electronic, mechanical, photocopying, microfilming, recordingor otherwise, without written permission from the Publisher, with the exceptionof any material supplied specifically for the purpose of being enteredand executed on a computer system, for exclusive use by the purchaser of the work.

Printed on acid-free paper

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To the authors’ families

Preface

Modern RF receivers and transmitters require quadrature oscillators with accuratequadrature and low phase-noise. Existing literature is dedicated mainly to singleoscillators, and is strongly biased towards LC oscillators. This book is devoted toquadrature oscillators and presents a detailed comparative study of LC and RC oscil-lators, both at architectural and at circuit levels. It is shown that in cross-coupled RCoscillators both the quadrature error and phase-noise are reduced, whereas in LC os-cillators the coupling decreases the quadrature error, but increases the phase-noise.Thus, quadrature RC oscillators can be a practical alternative to LC oscillators, es-pecially 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 two-integrator oscillators. The effect of mismatches on the phase-error and the phase-noise 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, withideal 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 topost-graduate students and lecturers, this book will be of interest to design engineersand researchers in this area.

The book originated from the PhD work of the first author. This work was thecontinuation of previous research work by the authors from TUDelft and Universityof 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 Tech-nical University of Lisbon), but part of the PhD work was done at TUDelft and at theUniversity of Alberta. This has influenced the work, by combining different viewsand backgrounds.

This book includes many original research results that have been presented atinternational conferences (ISCAS 2003, 2004, 2005, 2006, 2007 among others) andpublished in the IEEE Transactions on Circuits and Systems.

vii

viii Preface

Lisbon, Portugal Luis B. OliveiraLisbon, Portugal Jorge R.FernandesCanada Igor M. FilanovskyThe Netherlands Chris J.M. VerhoevenLisbon, Portugal Manuel M. Silva

Acknowledgements

The work reported in this book benefited from contributions from many personsand was supported by different institutions. The authors would like to thank allcolleagues at INESC-ID Lisboa, Delft University of Technology, and Universityof Alberta, particularly Chris van den Bos and Ahmed Allam, for their contribu-tions to the work presented in this book and for their friendly and always helpfulcooperation.

The authors acknowledge the support given by the following institutions:

� Fundacao para a Ciencia e Tecnologia of Ministerio da Ciencia, Tecnologia eEnsino 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.

� INESC-ID Lisboa (Instituto de Engenharia de Sistemas e Computadores –Investigacao e Desenvolvimento em Lisboa), Delft University of Technology,and University of Alberta, for providing access to their integrated circuit designand 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Organization of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Main Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Transceiver Architectures and RF Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Receiver Architectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Heterodyne or IF Receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.2 Homodyne or Zero-IF Receivers . . . . . . . . . . . . . . . . . . . . . . . . 102.2.3 Low-IF Receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Transmitter Architectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.1 Heterodyne Transmitters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.2 Direct Upconversion Transmitters . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4.1 Barkhausen Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4.2 Phase-Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4.3 Examples of Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5 Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.5.1 Performance Parameters of Mixers . . . . . . . . . . . . . . . . . . . . . . 272.5.2 Different Types of Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.6 Quadrature Signal Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.6.1 RC-CR Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.6.2 Frequency Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.6.3 Havens’ Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 Quadrature Relaxation Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 Relaxation Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.1 High Level Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2.2 Circuit Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3 Quadrature Relaxation Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.3.1 High Level Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

xi

xii Contents

3.3.2 Quadrature Relaxation Oscillator without Mismatches . . . . . . 423.3.3 Quadrature Relaxation Oscillator with Mismatches . . . . . . . . . 463.3.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.4 Phase-Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.4.1 Phase-noise in a Single Relaxation Oscillator . . . . . . . . . . . . . 563.4.2 Phase-noise in Quadrature Relaxation Oscillators . . . . . . . . . . 60

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4 Quadrature Oscillator-Mixer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.2 High Level Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2.1 Ideal Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.2.2 Effect of Mismatches and Delay . . . . . . . . . . . . . . . . . . . . . . . . 67

4.3 Circuit Level Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5 Quadrature LC-Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.2 Single LC Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.3 Quadrature LC Oscillator Without Mismatches . . . . . . . . . . . . . . . . . . 855.4 Quadrature LC Oscillator with Mismatches . . . . . . . . . . . . . . . . . . . . . . 895.5 Q and Phase-Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.6 Quadrature LC Oscillator-Mixer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6 Two-Integrator Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.2 High Level Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.2.1 Non-Linear Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006.2.2 Quasi-Linear Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.3 Circuit Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.4 Phase-Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.6 Two-Integrator Oscillator-Mixer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.6.1 High Level Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.6.2 Circuit Implementation and Simulations . . . . . . . . . . . . . . . . . . 114

6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

7 Measurement Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1197.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1197.2 Quadrature Relaxation Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7.2.1 Circuit Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1207.2.2 Measurement Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

7.3 Quadrature LC Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237.3.1 Circuit Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Contents xiii

7.3.2 Measurement Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1267.4 Quadrature Oscillator-Mixer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

7.4.1 Circuit Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277.4.2 Measurement Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

7.5 Comparison of Quadrature LC and RC Oscillators . . . . . . . . . . . . . . . . 1327.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

8 Conclusions and Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1378.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1378.2 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

A Test-Circuits and Measurement Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141A.2 Quadrature RC and LC Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

A.2.1 Test Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141A.2.2 Measurement Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

A.3 Quadrature Relaxation Oscillator-Mixer . . . . . . . . . . . . . . . . . . . . . . . . 144A.3.1 Test Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144A.3.2 Measurement Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

Chapter 1Introduction

Contents

1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Organization of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Main Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1 Background and Motivation

The huge demand for wireless communications has led to new requirements forwireless transmitters and receivers. Compact circuits, with minimum area, are re-quired to reduce the equipment size and cost. Thus, we need a very high degree ofintegration, if possible a transceiver on one chip, either without or with a reducednumber of external components. In addition to area and cost, it is very important toreduce 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 ofhighly sophisticated modulation techniques, complex demodulation algorithms, er-ror detection and correction, and data encryption, leading to a large improvementin the communication quality. Since digital signals are easier to process than ana-logue signals, a strong effort is being made to minimize the analogue part of thetransceivers by moving as many blocks as possible to the digital domain.

The analogue front-end of a modern wireless communication system is respon-sible for the interface between the antenna and the digital part. The analogue front-end of a receiver is critical, the specifications of its blocks being more stringentthan 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, butthe homodyne approach, after remaining a long time in the research domain, isbecoming a viable alternative [1, 2].

The main drawback of heterodyne receivers is that both the wanted signal and thedisturbances in the image frequency band are downconverted to the IF. Heterodynereceivers 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, suchfilters can only be implemented off chip (so far), and they are expensive. A highIF is required because, with a low-IF, the image frequency band is so close to thedesired frequency that an image-reject RF filter is not feasible.

Homodyne receivers do not suffer from the image problem, because the RFsignal is directly translated to the baseband (BB), without any IF. Thus, the maindrawbacks of the heterodyne approach (image interference and the use of externalfilters) are overcome, allowing a highly integrated, low area, low power, and lowcost receiver. However, homodyne receivers are very sensitive to parasitic basebanddisturbances 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 combi-nation with additive noise increases the bit error rate (BER).

A very interesting receiver approach, which combines the best features of thehomodyne and of the heterodyne receivers, is the low IF receiver [3–7]. This isbasically a heterodyne receiver using special mixing circuits that cancel the imagefrequency. Since image reject filters are not required, there is the possibility of usinga low IF, allowing the integration of the whole system on a single chip [4]. Thelow-IF receiver relaxes the IF channel select filter specifications, because it worksat a relatively low frequency and can be integrated on-chip, sometimes digitally. Theimage rejection is dependent on the quality of image-reject mixing, which dependson component matching and LO (local oscillator) quadrature accuracy. Thus, veryaccurate quadrature oscillators are essential for low-IF receivers.

Conventional heterodyne structures, with high IF, make the analogue to digitalconverter (A/D) specifications very difficult to fulfil with reasonable power con-sumption; therefore, the conversion to baseband has to be done in the analoguedomain. In the low-IF architecture, the two down converted signals are digitizedand mixed digitally to obtain the baseband as shown in Fig. 1.1.

Analogue Digital

LNA

I Q

LO1LO2

I Q

LPF

LPF A/D

A/D

BBI

BBQ

LNA - Low noise amplifierLPF - Low pass filter

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

1.1 Background and Motivation 3

The LO is a key element in the design of RF frontends. The oscillator should befully 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 veryimportant requirement of quadrature oscillators.

The most often used circuits to obtain two signals in quadrature have open-loopstructures, in which the errors are propagated directly to the output. Examples ofsuch structures are [1]:

1. Passive circuits to produce the phase-shift (RC-CR network), in which the phasedifference and gain are frequency dependent.

2. Oscillators working at double of the required frequency, followed by a divider bytwo; this method leads to high power consumption, and reduces the maximumachievable frequency.

3. An integrator with the in-phase signal at the input, followed by a comparator toobtain the signal in quadrature (aligned with the zero crossings of the integratoroutput); this has the disadvantage that the two signal paths are different.

In recent years, significant effort has been invested in the study of oscillators withaccurate quadrature outputs [9–11]. Relaxation and LC oscillators, when cross-coupled (using feedback structures), are able to provide quadrature outputs. In thisbook these oscillators are studied in depth, in order to understand their key parame-ters, such as phase-noise and quadrature error.

The relaxation oscillator has been somewhat neglected with respect to the LCoscillator, as it is widely considered as a lower performance oscillator in terms ofphase-noise. Although this is true for a single oscillator, it is not for cross-coupledoscillators.

In this work we consider alternatives to the LC oscillator and investigate theiradvantages and limitations. We study in detail the quadrature relaxation oscillatorsin terms of their key parameters, showing that due to the cross-coupling it is possibleto reduce the oscillator phase-noise and make the effect of mismatches a secondorder effect, thus improving the accuracy of the quadrature relationship. We showthat, 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 thirdtype of oscillator: the two-integrator oscillator. While in the previous cases we hadtwo oscillators with coupling to provide quadrature outputs, this oscillator is ableto provide inherent quadrature outputs, with phase-noise comparable to that of arelaxation 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 downconver-sion, with a direct influence on the global performance of the transceiver [1,2]. Theyhave been realized as independent circuits from the oscillators, either in heterodyneor homodyne structures. The evolution of mixer circuits has been, so far, essen-tially due to technological advancements in the semiconductor industry. Here, weshow that it is possible to integrate the mixing function with the quadrature oscilla-tors. 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 oscillatorsand mixers. We study the influence of the mixing function on the oscillator perfor-mance, and we confirm by measurement the oscillator-mixer concept. However, themain emphasis of this book is on the oscillators: the inclusion of the mixing functionstill requires further study.

In this work we study in detail the three types of quadrature oscillators referredabove, and we evaluate their relative advantages and disadvantages. Simulation andexperimental 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 asurvey, in Chapter 2, of RF front-ends and their main blocks: we describe the basicreceiver and transmitter architectures, then we focus on basic aspects of oscillatorsand mixers, and, finally, we review conventional techniques to generate quadraturesignals.

In Chapter 3 we present a study of the quadrature relaxation oscillator, in whichwe consider their key parameters: oscillation frequency, signal amplitude, quadra-ture relationship, and phase-noise. We use a structured approach, starting by con-sidering the oscillator at a high level, using ideal blocks, and then we proceed tothe analysis at circuit level. We present simulation results to confirm the theoreticalanalysis.

In Chapter 4 we analyse the quadrature relaxation oscillator-mixer. We first eval-uate the circuit at a high level (structured approach), deriving equations for the os-cillation frequency and quadrature error of the oscillator-mixer. We show that wecan inject the modulating signal in the circuit feedback loop, and we explain whereand 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 oscillationfrequency, signal amplitude, Q, and phase-noise. We investigate the possibility ofinjecting a signal to perform the mixing function.

In Chapter 6 we study the two-integrator oscillator. We proceed from a high leveldescription to the circuit implementation, and we present simulation results. We alsoshow the possibility of performing the mixing function in this oscillator.

In Chapter 7 we present several circuit implementations to provide experimentalconfirmation 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 suit-

able 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 internationalconferences and journals. It is believed that the main original contributions of thework are:

(i) A study (in Chapter 3) of cross-coupled relaxation oscillators using a structureddesign approach: first with ideal blocks, and then at circuit level. Equations arederived for the oscillation frequency, amplitude, phase-noise, and quadraturerelationship [12–15]. A prototype at 2.4 GHz was designed to confirm the maintheoretical results (quadrature relationship and phase-noise).

(ii) A study of a cross-coupled relaxation oscillator-mixer at high level (in chap-ter 4) [12, 16–18] and investigation of the influence of the mixing function onthe oscillator performance. A 5 GHz prototype was designed to validate theoscillator-mixer concept [19].

(iii) A study of cross-coupled LC oscillators concerning Q and phase-noise (inChapter 5) [20,21]. A comparative study of phase-noise in cross-coupled oscil-lators, which shows that coupled relaxation oscillators can be a good alternativeto coupled LC oscillators [14]. A 1 GHz prototype confirms the increase ofphase-noise in LC oscillators due to coupling [21], and two circuit prototypesat 5 GHz (RC and LC) confirm that quadrature RC oscillators might be a goodalternative to quadrature LC oscillators.

A minor contribution is the study of the two-integrator oscillator at high level and atcircuit level (in Chapter 6), in which we show that this circuit has the advantage ofa 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 ona relaxation oscillator has been proposed recently [26]. These results, however, areoutside of the scope of this book.

Chapter 2Transceiver Architectures and RF Blocks

Contents

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Receiver Architectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Heterodyne or IF Receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.2 Homodyne or Zero-IF Receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.3 Low-IF Receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Transmitter Architectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.1 Heterodyne Transmitters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.2 Direct Upconversion Transmitters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4.1 Barkhausen Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4.2 Phase-Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4.3 Examples of Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5 Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.5.1 Performance Parameters of Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.5.2 Different Types of Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.6 Quadrature Signal Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.6.1 RC-CR Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.6.2 Frequency Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.6.3 Havens’ Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.1 Introduction

In this chapter we review the basic transceiver (transmitter and receiver) architec-tures, and some important front-end blocks, namely oscillators and mixers. We givespecial attention to the conventional methods to generate quadrature signals.

We start by describing the advantages and disadvantages of several receiver andtransmitter architectures. Receivers are used to perform low-noise amplification,downconversion, and demodulation, while transmitters perform modulation, up-conversion, and power amplification. Receiver and transmitter architectures can bedivided into two types: heterodyne, which uses one or more IFs (intermediate fre-quencies), and homodyne, without IF. Nowadays research is more active concerningthe receiver path, since requirements such as integrability, interference rejection, and


7

8 2 Transceiver Architectures and RF Blocks

band selectivity are more demanding in receivers than in transmitters. The impor-tance of accurate quadrature signals to realize integrated receivers is emphasized inthis chapter.

At block level, the basic aspects of oscillators are reviewed, with special em-phasis on the phase-noise and its importance in telecommunication systems. Theoscillators can be divided into two main groups, according to whether they havestrong non-linear or quasi-linear behaviour. We present an example of each: the RCrelaxation oscillator (non-linear) and the LC oscillator (linear).

We survey the main characteristics of mixers, which, being responsible for fre-quency translation (upconversion or downconversion), are essential blocks of RFtransceivers.

This chapter ends by discussing the conventional methods to generate quadra-ture outputs, all of which employ open-loop structures. We describe in detail themost widely known method, the RC-CR network, and we also discuss two otherapproaches 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 thehomodyne and conventional IF architectures.

2.2.1 Heterodyne or IF Receivers

The heterodyne receiver was called by Armstrong as superheterodyne (patented in1917), because the designation heterodyne had already been applied in a differentcontext (in the area of rotating machines) [2]. This is the reason why the designationsuperheterodyne, instead of heterodyne, became prevalent until recently.

The heterodyne receiver has been, for a long time (more than 70 years), the mostcommonly used receiver topology. In this approach the desired signal is downcon-verted from its carrier frequency to an intermediate frequency (single IF); in somecases, it is further downconverted (multi IF). The schematic of a modern IF re-ceiver 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, anduses several discrete component filters. These filters must be implemented off-chip,with discrete components, to achieve high Q, which is difficult or impossible toobtain with integrated components. Using these high Q components, the heterodyne

2.2 Receiver Architectures 9

RFBPF

LNAIR

BPF

LO1

CSBPF

IF DSP

A/D

A/D

LO2

–90º

LPF

LPFImageReject

ChannelSelect

Fig. 2.1 Heterodyne receiver

receiver achieves high performance with respect to selectivity and sensitivity, whencompared with other receiver approaches [1, 4]. This receiver can handle modernmodulation schemes, which require the separation of I and Q signals to fully recoverthe information (for example, quadrature amplitude modulation); accurate quadra-ture outputs are necessary (for conversion to the baseband).

The main drawback of this receiver is that two input frequencies can produce thesame IF. For example, let us consider that the IF is 50 MHz and we want to down-convert a signal at 850 MHz. If we consider a local oscillator with 900 MHz, a signalat 950 MHz will be also downconverted by the mixer to the IF. This unwanted signalis called image. To overcome this problem in conventional heterodyne receivers animage 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 iseasier to design the image reject filter and suppress the image. However, in additionto the image, we also need to take into account interferers. At the IF frequency wemust remove interferers (which are also downconverted to the IF) using a channelselect filter (Fig. 2.1). Using a low IF reduces the demand on the channel selectfilter. Furthermore, a low IF relaxes the requirements on IF amplifiers, and makesthe A/D specifications easier to fulfil. Thus, there is a trade-off in the heterodynereceiver: with high IF image rejection is easier, whereas with low IF the suppressionof interferers is easier.

The heterodyne architecture described above requires the use of external com-ponents. It is not a good solution for low-cost, low area, and ultra compact modernapplications. The challenge nowadays is to obtain a fully integrated receiver, on a

ω1 ωLO ωIM ωIF

ωIF

0 ωω

ChannelChannel

Image RejectBPF

Image(rejectedby filter)

Image

Fig. 2.2 Image rejection


single chip. This requires either direct conversion to the baseband, or the develop-ment of new techniques to reject the image without the use of external filters. Thesetwo 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 singledownconversion (the IF is zero). These receivers, also called “direct-conversion”or “zero-IF”, are the most natural solution to detect information associated with acarrier in just one conversion stage. The resulting baseband signal is then filteredwith 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 approachimplies that the desired channel is its own image. Thus, the homodyne receiver doesnot require image rejection. All processing is performed at the baseband, and wehave the more relaxed possible requirements for filters and A/Ds.

Using modern modulation schemes, the signal has information in the phase andamplitude, and the downconversion requires accurate quadrature signals. The blockdiagram of a homodyne receiver is shown in Fig. 2.3. The filter before the LNAis optional [27], but it is often used to suppress the noise and interference outsidethe receiver band. This simple approach permits a highly integrated, low area, lowpower, and low-cost realization.

Direct conversion receivers have several disadvantages with respect to hetero-dyne receivers, which do not allow the use of this architecture in more demandingapplications. 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 closeto 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).

DSP

A/D

A/D

RFBPF LNA LO

–90º

LPF

LPF

Fig. 2.3 Homodyne receiver


(b) Channel selection – At the baseband the desired signal must be filtered, ampli-fied, and converted to the digital domain. The low-pass filter must suppress theout-of-channel interferers. The filter is difficult to implement, since it must havelow-noise and high linearity.

(c) LO leakage – LO signal coupled to the antenna will be radiated, and it will inter-fere with other receivers using the same wireless standard. In order to minimizethis effect, it is important to use differential LO and mixer outputs to cancelcommon mode components.

(d) Quadrature error – Quadrature error and mismatches between the amplitudesof the I and Q signals corrupt the downconverted signal constellation (e.g., inQAM). This is the most critical aspect of direct-conversion receivers, becausemodern wireless applications have different information in I and Q signals, andit is difficult to implement accurate high frequency blocks with very accuratequadrature relationship.

(e) DC offsets – Since the downconverted band extends down to zero frequency, anyoffset voltage can corrupt the signal and saturate the receiver’s baseband outputstages. Hence, DC offset removal or cancellation is required in direct-conversionreceivers.

(f) Intermodulation – Even order distortion produces a DC offset, which is signaldependent. Thus, these receivers must have a very high IIP2 (input second har-monic intercept point)

The direct conversion approach requires very linear LNAs and mixers, high fre-quency local oscillators with precise quadrature, and use of a method for achiev-ing submicrovolt offset and 1/f noise. All these requirements are difficult to fulfillsimultaneously.

2.2.3 Low-IF Receivers

Heterodyne receivers have important limitations due to the use of external im-age reject filters. Homodyne receivers have some drawbacks because the signalis translated directly to the baseband. Thus, there is interest in the developmentof new techniques to reject the image without using filters. An architecture thatcombines the advantages of both the IF and the zero-IF receivers is the low-IFarchitecture.

The low-IF receiver is a heterodyne receiver that uses special mixing circuitsthat cancel the image frequency. A high quality image reject filter is not necessaryanymore, 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, allowingthe integration of the whole system on a single chip. The low IF relaxes the IFchannel 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 IFis once to twice the bandwidth of the wanted signal. For example, an IF frequency


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 IFreceivers they have an additional use: accurate quadrature signals are essential toremove the image signal. This removal depends strongly on component matchingand LO (local oscillator) quadrature accuracy.

Two image reject mixing techniques can be used, which have been proposed byHartley and by Weaver.

The Hartley architecture [28] has the block diagram represented in Fig. 2.4. TheRF signal is first mixed with the quadrature outputs of the local oscillator. Afterlow-pass filtering of both mixers’ outputs, one of the resulting signals is shifted by90◦, and a subtraction is performed, as shown in Fig. 2.4. In order to show how theimage is canceled we must consider the signals at points 1, 2, and 3 (Fig. 2.4).

We assume that

xRF(t) = VRF cos(�RFt) + VIM cos(�IMt) (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

2sin[(�LO − �RF)t] + VIM

2sin[(�LO − �IM)t] (2.2)

x2(t) = VRF

2cos[(�LO − �RF)t] + VIM

2cos[(�LO − �IM)t] (2.3)

Equation (2.2) can be written as:

x1(t) = − VRF

2sin[(�RF − �LO)t] + VIM

2sin[(�LO − �IM)t] (2.4)

LO

–90°

LPF

cos (ω LO t)

sin (ω LO t)

LPF

RFInput

90°

IFOutput

1 3

2

Fig. 2.4 Hartley architecture (single output)


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

x3(t) = VRF

2cos[(�RF − �LO)t] − VIM

2cos[(�LO − �IM)t] (2.5)

Adding (2.5) and (2.3) cancels the image band and yields the desired signal. InHartley’s approach, the quadrature downconversion followed by a 90◦ phase shiftproduces in the two paths the same polarities for the desired signal, and oppositepolarities for image. The main drawback of this architecture is that the receiver isvery sensitive to the local oscillator quadrature errors and to mismatches in the twosignal paths, which cause incomplete image cancellation.

The relationship between the image average power (PIM) and the signal averagepower (PS) is [1]:

PIM

PS= V 2

IM

V 2RF

(VLO + �VLO)2 − 2VLO(VLO + �VLO) cos(�) + V 2LO

(VLO + �VLO)2 + 2VLO(VLO + �VLO) cos(�) + V 2LO

(2.6)

where VLO is the amplitude of local oscillator, �VLO is the amplitude mismatch, and� is the quadrature error.

Noting that VIM2/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 byVIM

2/VRF2.

IRR =PIM

PS

∣∣∣∣out

V 2IM

V 2RF

∣∣∣∣in

(2.7)

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

IRR ≈

(�VLO

VLO

)2

+ �2

4(2.8)

Note that we have considered only errors of the amplitude and phase in the localoscillator. Mismatches in mixers, filters, adders, and phase shifter will also con-tribute to the IRR. In integrated circuits, without using calibration techniques, thetypical values for amplitude mismatch are 0.2–0.6 dB and for the quadrature error3−5◦, 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 inboth signal paths: the second stage of I and Q mixing has the same effect of the90◦ phase shift used in the Hartley approach. As with the Hartley receiver, if the


RFInput

LPF

LPF

IF or BBOutput

LO1 LO2

–90°–90°

Fig. 2.5 Weaver architecture with single output

phase difference of the two local oscillator signals is not exactly 90◦, the image isno longer completely cancelled.

The Weaver architecture has the advantage that the RC-CR mismatch effect (thiseffect will be discussed in detail in Section 2.6) on the 90◦ phase shift after the down-conversion in the Hartley architecture is avoided and the second order distortion in

ωLO1

ωLO2

2ωLO2 – ωIN + 2ωLO1

2ωLO2 – ωIN + ωLO1

ωIN ω

ω

ω

ωIN – ωLO1

ωIN – ωLO1 – ωLO2

REInput

Channel

Channel

Channel

SecondaryImage

SecondaryImage

SecondaryImageSecond

IF

FirstIF

Fig. 2.6 Secondary image problem in the Weaver architecture

2.3 Transmitter Architectures 15

I Q

LO1

I Q

BB I

BB Q

LO2

RFInput

LPF

LPF

LPF

LPF

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 inamplitude and quadrature error of the two LO signals. It suffers from an imageproblem (in the second mixing operation) if the second downconversion is not tothe baseband, as shown in Fig. 2.6. In this case the low pass filters, must be replacedby bandpass filters, to suppress the secondary image, but, the image suppression iseasier at IF than at RF.

The receiver of Fig. 2.5 needs to be modified to provide baseband quadratureoutputs, which are necessary in modern wireless applications. We need 6 mixersto cancel the image and separate the quadrature signals, as shown in Fig. 2.7. Thesecond two mixers in Fig. 2.5 are replaced by two pairs of quadrature mixers, andtheir outputs are then properly combined [2]. This modified Weaver architecture isused 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 archi-tecture in transmitters. In heterodyne transmitters the baseband signals are modu-lated 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 filterthat follows rejects the harmonics of the IF signal, and reduces the transmitted noise.


LO

RFBPF PA

IFBPFDSP

D/A

D/A

LO

–90°

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 emissionlevels imposed by the standards. This filter is typically passive and built with expen-sive off-chip components [1, 4]. This topology does not allow full integration of thetransmitter, 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 upcon-verted to RF. The RF carrier frequency is equal to the LO frequency, at the mixersinput. A quadrature upconversion is required by modern modulations schemes.

This topology can be easily integrated, because there is no need to suppressany mirror signal generated during the upconversion. As in the receiver, the localoscillator frequency is the carrier frequency [4].

HFBPFPADSP

D/A

D/A

LO

–90°

Fig. 2.9 Direct upconversion transmitter

2.4 Oscillators 17

The main disadvantage is the “injection pulling” or “injection locking” of thelocal oscillator by the high level PA output. The resulting spectrum can not besuppressed by a bandpass filter, because it has the same frequency as the wantedsignal. 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 bothdirect and heterodyne upconversion was proposed in [32, 33]. In this case the base-band signals are converted to a low IF, and are then upconverted to the final carrierfrequency using an image-reject mixing technique to reject the unwanted sideband,thus avoiding the use of an RF filter after the upconversion. Thus, an integratedcircuit realization is possible, with lower area and cost than with a conventionalheterodyne 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(�0t + �) (2.9)

For digital applications, oscillators generate a clock signal, which is a square-waveform with period T0.

Sinusoidal oscillators can be analyzed as a feedback system, shown in Fig. 2.11,with the transfer function.

Yout ( j�)

Xin( j�)= H ( j�)

1 − H ( j�)�( j�)(2.10)

Amplitude [V]

(a) (b)

ω0 ω[rad/s]

P[dBm]

t[s]

V0

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


Fig. 2.11 Feedback systemblock diagram

+ ++

Xin ( jω) Yout ( jω)H( jω)

β( jω)

The necessary conditions concerning the loop gain for steady-state oscillationwith frequency �0 are known as the Barkhausen conditions. The loop gain must beunity (gain condition), and the open-loop phase shift must be 2k�, where k is aninteger 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 systemis 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 idealand real oscillators is the phase-noise. The noise generated at the oscillator out-put causes random fluctuation of the output amplitude and phase. This meansthat 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 sidebandsdecreases [35].

The noise can be generated either inside the circuit (due to active and passivedevices) or outside (e.g., power supply). Effects such as nonlinearity and periodic

Fig. 2.12 Spectrum ofoscillator output withphase-noise

P [dBm]

ω[rad/s]ω0 2ω0

White noisefloor

3ω0

2.4 Oscillators 19

variation of circuit parameters make it very difficult to predict phase-noise [36]. Thenoise causes fluctuations of both amplitude and phase. Since, in practical oscillatorsthere is an amplitude stabilization scheme, which attenuates amplitude variations,phase-noise is usually dominant. The oscillator noise can be characterized eitherin the frequency domain (phase-noise), or in the time domain (jitter). The firstis 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 inoscillators (a review of different standards and measurement methods is presentedin [38]). They are often characterized in terms of the single sideband noise spec-tral density, L(�), expressed in decibels below the carrier per hertz (dBc/Hz). Thischaracterization 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 doneby using a spectrum analyzer (which is a general-purpose equipment, but will in-troduce some errors) or with phase or frequency demodulators with well knownproperties (which are dedicated and expensive equipment). Note that the spectraldensity (2.13) includes both phase and amplitude noise, and they can not be sep-arated. 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 simplyas “phase-noise”.

The carrier-to-noise ratio (CNR) can also be used to specify the oscillator phase-noise. 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 isrelated to the total oscillator phase-noise. Q is usually defined within the context ofsecond order systems. There are three possible definitions of Q [1]:

(1) Leeson in [39] considers a single resonator network with −3 dB bandwidthB and resonance frequency �0 (Fig. 2.13),


Fig. 2.13 Q definition for asecond order system

ω0 ω

B

3 dB

H(s)

Q = �0

B(2.15)

A second order bandpass filter has a transfer function:

H (s) =K

�0

Qs

s2 + �0

Qs + �2

0

(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 thisapproximation is valid for Q ≥ 5. This definition of Q is suitable for filters, andcan be used in oscillators if we consider the resonator circuit as a second orderfilter.

(2) A second definition of Q considers a generic circuit and relates the maximumenergy 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 maxi-mum energy stored (in C or L) and the energy dissipated (by R) in a period. As anexample, we apply the definition to an RLC series circuit. The energy is stored inthe inductor and the capacitor, and the maximum energy stored in the inductor andthe capacitor is the same.

The energy stored in an inductor (WL) is:

WL =T∫

0

i(t)Ldi(t)

dtdt = L I 2

rms (2.18)

where Irms is the root-mean-square current in the inductor.The energy dissipated in a resistor (WR) per cycle (in the period T0) is:

WR = I 2rms RT0 (2.19)

2.4 Oscillators 21

The value of Q is:

Q = 2�L I 2

rms

I 2rms RT0

= 2� f0L

R= �0 L

R(2.20)

We can also use the energy stored in the capacitor:

WC =T∫

0

ν(t)Cdν(t)

dtdt = CV 2

rms (2.21)

where Vrms is the root-mean-square voltage in the capacitor, and Vrms = Irms/(�0C).Then, the value of Q is:

Q = 2�CV 2

rms

I 2rms RT0

= 2� f0(C I 2

rms)/(�20C2)

RI 2rms

= 1

�0C R(2.22)

Equation (2.17) is a general definition, which does not specify which elementsstore or dissipate energy.

(3) In the third definition of Q the oscillator is considered as a feedback systemand the phase of the open-loop transfer function H (j�) is evaluated at the oscillationfrequency, �osc, which is not necessarily the resonance frequency [36]. In a singleRLC circuit the oscillation frequency is the resonance frequency, but with coupledoscillators the oscillation frequency can be different, as we will show in Chapter 5.The oscillator Q is defined as:

Q = �0

2

√(

d A

d�

)2

+(

d�

d�

)2

(2.23)

where A is the amplitude and � is the phase of H (j�). This definition, called open-loop Q, was proposed in [36], and takes into account the amplitude and phase vari-ations 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 tocalculate the degradation of the oscillator quality factor if the oscillation frequency is

Fig. 2.14 Definition of Qbased on open-loop phaseslope

RCL

H( jω)

ω0 ω

θ = ∠H( jω)


different from the resonance frequency. We will also use this definition in Chapter 6to 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 empir-ical equation proposed in [39–41]. It is based on the assumption that the oscillatoris a linear time invariant system. The following equation for L(�m) is obtained [42]:

L(�m) = 10 log

{

2FkT

PS

[

1 +(

�0

2Q�m

)2](

1 + �1/ f 3

|�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 donein [43].

A different model to predict the oscillator phase-noise was presented recentlyin [42]. This is a linear time-variant model, which, according to the authors, givesaccurate 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]:

Fig. 2.15 A typicalasymptotic noise spectrum atthe oscillator output ω1 ω2 ωω0

–30 dB/decade

–20 dB/decade

white noise floor

(3) (2) (1)

(ω)(dBc/Hz)

2.4 Oscillators 23

(1) For frequencies far away from the carrier, the noise of the oscillator is due towhite-noise sources from circuits, such as buffers, which are connected to the oscil-lator, 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 byits 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 thedesired oscillation frequency. This phenomenon will limit the immunity against ad-jacent interferer signals: in the receiver path we want to downconvert a specificchannel located at a certain distance from the oscillator frequency; due to the os-cillator phase-noise, not only the desired channel is downconverted to an interme-diate frequency, but also the nearby channels or interferers, corrupting the wantedsignal [35] (Fig. 2.16). This effect is called “reciprocal mixing”.

In the case of the transmitter path the phase-noise tail of a strong transmittercan 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 transmittersignal at �1 with substantial phase-noise.

Fig. 2.16 Phase-noise effecton the receiver and theundesired downconversion ω ωωIFω0

Signal Signal

Interferer

Interferer

Fig. 2.17 Phase-noise effecton the transmitter path ωω1

CloseTransmitter

Signal

ω2


2.4.3 Examples of Oscillators

Oscillators can be divided into two main groups: quasi-linear and strongly non-linear oscillators [34].

Strongly non-linear or relaxation oscillators are usually realized by RC-activecircuits. In this book we will present a detailed study of relaxation oscillators. Themain advantage of this type of oscillators is that only resistors and capacitors areused together with the active devices (inductors, which are costly elements in termsof chip area, are not needed); the main drawback of relaxation oscillators is theirhigh phase-noise. In addition to the relaxation oscillator, another RC oscillator willbe studied: the two-integrator oscillator. This oscillator is very interesting because itcan 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 resonatorelement: dielectric resonators, crystals, striplines, and LC tanks. These oscillatorsare known by their good phase-noise performance, since Q is normally much higherthan one.

In this book we are interested in oscillators capable to produce two outputs inaccurate quadrature. We will study relaxation RC oscillators and LC oscillators withan LC tank (usually called simply LC oscillators), because they can be cross-coupledto provide quadrature outputs. We also study a third type of oscillator: the two-integrator oscillator, which has inherent quadrature outputs. In the next part of thissection 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 theydo not have inductors), in applications with relaxed phase-noise requirements [9],typically as part of a phase-locked loop. However, these oscillators have not beenpopular in RF design because they have noisy active and passive devices [1].

Fig. 2.18 Relaxationoscillator

VCC

RR

M M

CII

2.4 Oscillators 25

In Fig. 2.18 we present an example of an RC relaxation oscillator. This oscillatorhas been referred to as a first order oscillator, since its behaviour can be describedin terms of first order transients [8, 44]. It operates by alternately charging and dis-charging a capacitor between two threshold voltage levels that are set internally. Theoscillation frequency cannot be determined by the Barkhausen criterion (this is nota 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 becauseit is a quasi-linear oscillator. Oscillation will occur at the frequency for which theamplitude of the loop gain is one and the phase is zero. The LC oscillator modelis represented in Fig. 2.19: the transfer function is H ( j�) = gm and �( j�) is theimpedance of the parallel RLC circuit.

�( j�) = R

1 + j

(�

�0− �0

�

)

Q(2.25)

where

Q = R

√

C

L(2.26)

�0 = 1√LC

(2.27)

At the resonance frequency (�0) the inductor and capacitor admittances canceland the loop gain is |H (j�0)�( j�0)| = gm R = 1: the active circuit has a negativeresistance, which compensates the resistance of the parallel RLC circuit. This con-dition is necessary, but not sufficient, because, for the oscillation to start, the loopgain must be higher than 1, gm > 1/R.

In Fig. 2.20 a typical LC oscillator, used in RF transceivers, is shown. This isknown as LC oscillator with LC-tank, and it is also called differential CMOS LC

Fig. 2.19 LC-oscillatorbehavioural model

gm

C R L


Fig. 2.20 CMOS LCOscillator with LC tank

VCC

LL

M M

I

CC

Fig. 2.21 Equivalentresistance of thedifferential pair

–gm

2

vx

2

vx gm 2

vx2

vx

2

vx

ix

+vx–

–

2

vx–

oscillator, or negative gm oscillator. The cross-coupled NMOS transistors (M) gen-erate 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

ix= − 2

gm(2.28)

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

2.5 Mixers

Mixers are a fundamental block of RF front-ends. Nowadays, a research effort isdone to realize a fully integrated front-end, to obtain cost and space savings. Inte-grated 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 willinvestigate the combination of the oscillator and the mixer in a single block.

Conventional mixers have an open-loop structure, in which the output is obtainedby the multiplication of a local oscillator signal and an input signal (RF signal, inthe receiver path). For quadrature modulation and demodulation two independentmixers are required, which imposes severe constrains on the matching of circuitcomponents. The integration of the mixing function in quadrature oscillators has theadvantage 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 fig-ure, second and third order intermodulation points, 1-dB compression point, gain,input and output impedance, and isolation between ports. Different types of imple-mentations 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 theoutput. It is an important measure of the performance of the mixer, indicating howmuch noise is added by it. The noise factor of a noiseless system is unity, and it ishigher in real systems.

NF = SNRIN

SNROUT(2.29)

The intermodulation distortion (IMD) is a measure of the mixers linearity. In-termodulation distortion is the result of two or more signals interacting in a nonlinear device to produce additional unwanted signals. Two interacting signals willproduce intermodulation products at the sum and difference of integer multiples ofthe original frequencies.

For two input signals at frequencies f1 and f2, the output components will havefrequencies m f1 ± n f2, where m and n are integers. The second and third-orderintercept points (IP2 and IP3) can be defined for the input (IIP2 and IIP3), or forthe 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 inputpower. IIP3 and OIP3 are the input and output power, respectively, at the point ofintersection (extrapolated) of the two lines. The IIP3 can be determined for any inputpower (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 betweenthe 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 andoutput intercept points (IIP2 and OIP2), which are obtained from the intersectionpoint 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 isthe most important. If two input tones at f1+ fLO and f2+ fLO are close in frequency,the intermodulation components at 2 f2 − f1 and 2 f1 − f2 will be close to f1 and


f1 f2

2f1 − f2 2f2 − f1 f

(a) (b)

ΔP

2IIP3 = PIN (dBm)

PIN (dBm)

+ ΔP

POUT (dBm)

P1

P3

IIP3

ΔP

2ΔP

OIP3

IP3

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

f2, making them difficult to filter without also removing the desired signal. Higherorder intermodulation products are usually less important, because they have loweramplitudes, and are more widely spaced. The remaining third order products, 2 f1 +f2 and 2 f2 + f1, 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( f1 and f2), can be close to DC ( f2 − f1 and f1 − f2), and lie in the signal band(Fig. 2.23). Thus, a mixer that converts directly to the baseband has very stringentIP2 requirements.

Another specification concerning distortion is the 1-dB compression point. Thisis the output power when it is one dB less than the output power of an extrapolatedlinear 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 signalto the rms voltage of the RF signal.

Baseband

0

SignalSignal

RF

f2 − f

1f1 − f

2

f1

f2

fLO

fLO

fLO

f1 + f

LOf2 +

Fig. 2.23 Second order distortion in a direct conversion mixer

2.5 Mixers 29

Fig. 2.24 Calculationof P-1dB 1 dB

POUT (dB)

PIN (dB)P–1dB

Voltage Gain(dB) = 20 log

(VOUT

VIN

)

(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 thevoltage and power conversion gains are equal.

In conventional heterodyne receivers the input impedance of the mixer must be50 � because we need an external image reject filter, which should be terminated by50 � 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 needsto be matched to the LNA output.

The isolation between the mixer ports is critical. This quantifies the interactionamong the RF, IF (or baseband for homodyne receivers), and LO ports. The LO toRF 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 withthe local oscillator that drives the mixer. The LO to IF feedthrough is undesirablebecause 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 ofthe implementation is based on linearity, gain, and noise figure requirements. Thesimplest mixer is a switch, implemented by a CMOS transistor [1]. The circuit ofFig. 2.25 is referred to as a passive mixer; although having an active element, thetransistor, this acts as a switch, and does not provide gain. This type of mixers,


Fig. 2.25 Mixer using aswitch

vLO

vRF vIF

RL

typically has no DC consumption, has high linearity and high bandwidth, and issuitable 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 sub-sequent stages; they are referred to as active mixers. These are widely used in RFsystems, and most of them are based on the differential pair. They can be divided intosingle-balanced mixers, where the LO frequency is present in the output spectrumand double-balanced mixers, which use symmetry to remove the LO frequency fromthe output.

In the single-balanced mixer, the differential pair has the LO signal at the inputand the current source is controlled by the other input signal (RF signal in downcon-version, as shown in Fig. 2.26). It converts the RF input voltage to a current, whichis steered either to one or to the other side of the differential pair. This mixer hasthe advantage that it is simple to design and operates with a single-ended RF input.

Fig. 2.26 Activesingle-balanced mixer

M1 M2

M3

VCC

RR

vIF

vLO

vRF

2.6 Quadrature Signal Generation 31

Fig. 2.27 Activedouble-balanced mixer

M2

M5 M6

M1 M3 M4

RvIF

vLO

vRF

R

I

VCC

When compared with a double-balanced mixer, it has moderate gain and moderatenoise figure, low 1 dB compression point, low port-to-port isolation, low IIP3, andhigh 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 RFdifferential inputs: it is the Gilbert cell [1, 2], represented in Fig. 2.27. This mixerhas higher gain, lower noise figure, good linearity, high port-to-port isolation, highspurious rejection, and less even order distortion, with respect to the single balancedmixer. The main disadvantage is the increased area (due to complexity) and powerconsumption [1,46]; additionally, it may require a balun transformer [46] to providethe 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 demod-ulation and for image rejection. The common methods of generating signals with aphase difference of 90◦, employ open-loop structures [1], and are reviewed in thissection. 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 divi-sion 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 whichthe input is shifted by +45◦ in the CR branch and by −45◦ in the RC branch. Theoutputs 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 decreasesasymptotically to −90◦. The phase shift of vOUT2 is +90◦ at DC and decreases


Fig. 2.28 Quadraturegeneration using an RC-CRcircuit

C

R

vIN

vOUT1

vOUT2

R

C

with the frequency towards 0◦. The phase shift of each branch changes with thefrequency, but the phase difference of the two outputs is always 90◦. This approachprovides a good quadrature relationship, but the amplitude of the outputs changessignificantly with the frequency. The I and Q branches have, respectively, a low-passand a high-pass characteristic. The two output amplitudes are only equal at the polefrequency, �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, havinga direct influence on the value of the frequency at which there are quadrature sig-nals with equal amplitude. To minimize this problem, the amplitudes can be equal-ized by using limiter stages based on differential pairs [1] or using variable gainamplifiers [2].

At the pole frequency there is 3 dB attenuation, which is a significant loss. More-over, this network generates thermal noise, which can not be ignored.

In the circuit of Fig. 2.28, the mismatch of resistors and capacitors originates adeviation � from the 90◦ phase difference. Assuming relative mismatches � for theresistances and � for the capacitances, we can express � in the neighbourhood of� = 1/(RC) as:

� = �

2− {arctan[R(1 + �)C(1 + �)�] − arctan(RC�)} (2.32)

Using the trigonometric relationship

arctan(A) − arctan(B) = A − B

1 + AB(2.33)


we obtain

� = �

2− arctan

(RC�(1 + �)(1 + �) − (RC�)

1 + RC�(1 + �)(1 + �)RC�

)

(2.34)

If � � 1 and � � 1 (small mismatches), and taking into account that � ≈1/(RC)):

� ≈ �

2− arctan

(� + �

2

)

(2.35)

� ≈ �

2− � + �

2(2.36)

For typical values � = � = 10%, equation (2.36) gives 5.73◦ worst-case quadra-ture error.

An RC-CR network with two or more stages is known as a polyphase filter. A sin-gle RC-CR stage provides (without mismatches) an amplitude error below 0.2 dBover a 10% bandwidth. A properly designed 2-stage RC-CR network can give thesame gain error with a higher bandwidth. We can use more stages in order to coverthe required bandwidth. However, a polyphase filter has significant attenuation andhigh noise [2].

To avoid these problems other quadrature techniques may be used, which provideinherently quadrature outputs with equal amplitude.

2.6.2 Frequency Division

Another approach to generate quadrature carriers is frequency division. This is asimple technique in which a master-slave flip-flop is used to divide by two thefrequency of a signal with double of the desired frequency (Fig. 2.29). If vIN has50% duty-cycle, then the outputs are in quadrature [1].

The use of a carrier with twice the desired frequency has two main disadvan-tages: there is an increase in the power consumption, and the maximum achiev-able frequency is reduced. Mismatches in the signal paths through the latches and

Fig. 2.29 Frequency divideras a quadrature generator

vOUT1

Latch

Latch

vINvOUT2


deviations of the input duty-cycle from 50% contribute to the phase error. In orderto reduce the quadrature error, two dividers can be used, but this requires an inputsignal 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 byusing 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 afterthe phase shifter (RC-CR network). After this limiting action, the two signals areadded and subtracted, and the results are again limited, generating the two finaloutputs, which are approximately sinusoidal (since the limiter is “soft”) and inquadrature:

v1(t) + v2(t) = 2A cos�

2cos

(

�t + �

2

)

(2.39)

v1(t) − v2(t) = 2A sin�

2sin

(

�t + �

2

)

(2.40)

v1

v2

vOUT1

vOUT2

(a) (b)

vIN

Soft-limiter

~90°

v2

v1

vOUT1

vOUT2

–v2

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


The main advantage of this approach is that, although any error in the 90◦ phaseshift block leads to an amplitude mismatch between the two outputs, this is cancelledby the soft-limiters, as shown in equations (2.39) and (2.40).

This method is robust with respect to amplitude errors; however, the need offour soft-limiters and two adders makes this circuit less attractive for low-power,low area, and low cost applications. The above analysis assumes quasi-sinusoidalsignals. However, the soft-limiters and the non-linearity of the adders generate har-monics. This is an important drawback of this approach: even order harmonics with90◦ phase difference results in quadrature errors, and odd order harmonics produceamplitude mismatch. Finally, the capacitive coupling between the inputs of the twoadders is an extra source of quadrature error [1].

It is important to note that in the Havens technique the generated quadraturesignals are usually quasi-sinusoidal, while in the frequency division approach theoutputs have a square waveform.

All the conventional quadrature generating circuits, reviewed above, have open-loop architectures, in which the errors are propagated to the output. In this book, wewill study closed-loop architectures, which have better quadrature accuracy.

Chapter 3Quadrature Relaxation Oscillator

Contents

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 Relaxation Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.1 High Level Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2.2 Circuit Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3 Quadrature Relaxation Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.3.1 High Level Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.3.2 Quadrature Relaxation Oscillator without Mismatches . . . . . . . . . . . . . . . . . . . . 423.3.3 Quadrature Relaxation Oscillator with Mismatches . . . . . . . . . . . . . . . . . . . . . . . 463.3.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.4 Phase-Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.4.1 Phase-noise in a Single Relaxation Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.4.2 Phase-noise in Quadrature Relaxation Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.1 Introduction

This chapter is dedicated to the study of quadrature relaxation oscillators, whichconsist of two cross-coupled RC relaxation oscillators [9]. In this book we will useinterchangeably the designations cross-coupled relaxation oscillator and quadraturerelaxation oscillator.

Both single relaxation oscillators and the technique of synchronously couplingrelaxation oscillators have been known for some time [8–10, 44, 47–51], but theirresearch is still at an initial stage. We present a detailed study of the cross-coupledoscillator using a structured design approach: we first represent the oscillator at ahigh 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 thebasic aspects of single relaxation oscillators and of cross-coupled relaxation oscil-lators. We present a detailed study of the effect of mismatches on the output voltageand period of oscillation, and we calculate the quadrature error. This analysis isrigorous for low frequency. At high frequency several other effects exist, which arevery difficult to include in simple and tractable equations. Thus, at high frequency


37

38 3 Quadrature Relaxation Oscillator

we only show how the quadrature relaxation oscillator will react to mismatches (bychanging the amplitudes) in order to preserve an accurate quadrature relationship.

The second part of this chapter is dedicated to the study of the oscillator phase-noise. We identify the oscillator noise sources and we analyse the phase-noise ofsingle relaxation oscillators. The analysis of cross-coupled oscillators is rather com-plicated, but a simple qualitative argument indicates that the coupling reduces thephase-noise (as opposed to what happens in coupled LC oscillators). This is demon-strated 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 mod-elled using an integrator and a Schmitt trigger. The Schmitt-trigger is a memoryelement, and controls (switches) the sign of the integration constant. The oscillatorwaveforms are presented in Fig. 3.1b: the square waveform is the Schmitt-triggeroutput, and the triangular waveform is the integrator output.

Integrator

a)

b)

Schmitt-trigger

vINT vST

vST vINT

Am

plitu

de

Time

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 aspossible (Fig. 3.2). Thus, we should substitute the integrator and the Schmitt triggerby simple circuits that ensure some correspondence between the high level and thecircuit level. The integrator is implemented simply by a capacitor (Fig. 3.3); its inputis the capacitor current (iC) and the output is the capacitor voltage (νC). This voltageis the input of the Schmitt-trigger (Fig. 3.4), the output of which is iC. The transfercharacteristic of the Schmitt-trigger is shown is Fig. 3.4b. It is assumed that theswitching occurs abruptly when the sign of νBE1 − νBE2 changes.

Using this approach we have justified the implementation of the known circuitpresented 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 iC, it isconvenient to use as the oscillator output the voltage νOUT = ν1 − ν2 [8], with anamplitude of 4IR, as shown in Fig. 3.5. However, this is only valid for low oscillation

Fig. 3.2 Relaxation oscillatorimplementation, suitable forhigh frequency

R R

Q2

C

VCC

Q1

v1

vC

I I

v2

C

a) b)

vC

iCvC (V )

t(s)C

I

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


I I

i1 i2

R R

b)a)

Q2

VCC

Q1

v1 v2

vC

I

I−

RIiC = i1 – I 2− RI2

vC

iC+ –

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

Fig. 3.5 Relaxation oscillatorwaveforms

t(s)

–2IR

2IR

[V ]

Amplitude

ν OUTν INT

frequencies; at very high frequencies the outputs are approximately sinusoidal (theharmonics are filtered out by the circuit parasitics) with an amplitude lower than4IR. 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 circuitwith a more straightforward correspondence between the blocks and their circuitrealization was presented in [52]; however, in this case the maximum available fre-quency is reduced, and the noise, area, and power consumption are higher.

This oscillator integration constant is I/C, and the amplitude is 4IR. Thus, theoscillation frequency is:

f0 = I

2C(4RI )= 1

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 providequadrature outputs. If we add a soft-limiter (an amplifier with saturation) after theintegrator, as represented in Fig. 3.6a, we obtain a new output, with 90◦ phase differ-ence with respect to the Schmitt-trigger output (Fig. 3.6b). By increasing the limitergain this new output will be closer to a square signal with 90◦ phase shift (withinfinite 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 quadra-ture generator, but it has an open-loop structure, in which the output signals havedifferent paths, so there is an error in the quadrature relationship. However, the soft-limiter output can be used to synchronize a second relaxation oscillator. Couplingtwo oscillators using this technique leads to a cross-coupled relaxation oscillator ina feedback structure (Fig. 3.7) [8].

In the cross-coupled oscillator there is not a master and a slave oscillator: bothoscillators trigger each other in a balanced structure. The two oscillators lock at

Fig. 3.6 (a) Relaxationoscillator and a soft-limiter.(b) Soft-limiter input andoutput

vSL

vSL

vINT

vINT

Timeb)

Integrator Schmitt-trigger

Soft-limiter

a)

Amplitude


Fig. 3.7 Cross-coupledrelaxation oscillator

1STv

ST2v

1v1INTv

INT2v

1SLv

SL2v

2v

Integrator

Integrator

1

1

2

2

2

1

Soft-limiter

Soft-limiter

Schmitt-trigger

Schmitt-trigger

the same frequency, and the two Schmitt-trigger outputs have inherently 90◦ phasedifference (Fig. 3.8a). Figure 3.8b shows the effect of adding the soft-limiter outputof one oscillator to the integrator triangular signal of the other. The transition ofeach oscillator is now defined by a signal with a steeper slope, which means thatthe switching time is less sensitive to noise. Thus, to minimize the influence ofnoise on the transition time we should increase the soft-limiter gain. Note that whenthe two oscillators are equal, the amplitudes and the frequencies of each individualrelaxation 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 oscilla-tors, which are cross-coupled using, as coupling blocks (soft-limiters), differentialpairs that sense the capacitor voltage and have the differential output connected tothe other oscillator, as shown in Fig. 3.9 [8]. The soft-limiter output is a differentialcurrent, which is added at the collector nodes. The effect of this current is to changethe input switching levels of the Schmitt-trigger. Thus, this is equivalent to addinga voltage signal at the Schmitt-trigger input, as indicated in the block-diagram ofFig. 3.7.

In this section we assume that the coupled oscillators are equal, without mis-matches: C1 = C2 = C , I1D = I1C = I2D = I2C = I , and ISL1 = ISL2 = ISL . Eachof the coupled oscillators in Fig. 3.9 can be studied as a relaxation oscillator with twoextra current sources, which are responsible for the coupling action. For instance,the two current sources iSL1 and iSL2 are provided by the soft-limiter circuit drivenby the second oscillator, i.e., one oscillator is synchronously switched (triggered) bythe other oscillator.


Fig. 3.8 (a) Integratoroutput; (b) Schmitt-triggerinput with a steeper slope inthe transition region

Time

Timea)

Timeb)

1STv

2STv

INT1v

1v

Am

plitu

deA

mpl

itude

Am

plitu

de

We assume that there are no mismatches between the two oscillators and that theswitching occurs instantly when there is a transition of the capacitor voltages byzero. The transistors are assumed to act as switches, which is a good approximationfor bipolar transistors; this is also valid for MOS implementations (with high W/Ltransistors). The waveforms are shown in Fig. 3.10.

In this analysis we show why the introduction of coupling will not change theamplitudes of νC1 and νC2 and the oscillation frequency, with respect to the iso-lated oscillators. In order to determine the amplitudes νC1 and νC2, we must de-termine their maximum and minimum values. Due to the oscillator symmetry wewill only do the calculations for νC1 (for the amplitude of νC2 the results are thesame).


R R

DI1

2Q

1C

CI1

CCV

1Q

1Cv

R R

DI2

4Q

2C

CI2

3Q

2Cv

SLQ SLQ SLQ SLQ

12 SLI 22 SLI

1v 2v 3v4v

1SLi2SLi 3SLi 4SLi

Fig. 3.9 Circuit implementation of a quadrature relaxation oscillator

To calculate the amplitude of νC1 we must consider the two extremes, at instantst1 and t3 in Fig. 3.10. We consider that the oscillator is in steady-state (we do notstudy the transient regime), and we assume that νC1 and νC2 are in quadrature; thewaveform in advance of 90◦ can be either νC1 or νC2, depending on which of the twocoupling connections is direct and which is reversed. Note that the oscillators canlock in phase or in quadrature, depending on the sign of the summations at the inputof the Schmitt-trigger (Fig. 3.7), i.e. the polarity of connection of the soft-limitersin 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 Q2 is on and Q1 is off. When νC2(t)

goes through zero (instant t1 in Fig. 3.10) iSL1 decreases and iSL2 increases. We have

{

ν1 = VCC − iSL1 R

ν2 = VCC − 2I R − iSL2 R(3.2)

The transistors change state when νB E1 = VBE O N , and immediately before theswitching occurs,

νC1 = ν2 − ν1 + νB E2O N − νB E1O N


Fig. 3.10 Waveforms in asymmetric quadratureoscillator (withoutmismatches)

1 C v

R I 2 −

2 C v

1 t 0 t 2 t 3 t 4 t

1 T 3 T 2 T 4 T

RI 2

C

I slope

RI 2

R I 2 −

R I 2 −

off 1 Q

on 3 Q

on 1 Q

on 3 Q off 1 Q

off 3 Q

on 1 Q off 3 Q

C

I slope

) ( 2 SL I I R +

) ( 2 SL I I R + −

) ( 2 SL I I R +

) ( 2 SL I I R + −

2 1 v v −

4 3 v v −

Assuming that νB E2O N ≈ νB E1O N ,

νC1 ≈ ν2−ν1 = VCC−2I R−iSL2 R−[VCC−iSL1 R] = −2I R−R(iSL2−iSL1) (3.3)

Since it is assumed that the switching is provided by a vanishingly small valueof iSL2 − iSL1 the minimum value of νC1 is −2I R.

Considering now the second transition of νC2 by zero (instant t3 in Fig. 3.10) iSL1

increases and iSL2 decreases. With Q1 on and Q2 off we have:


{

ν1 = VCC − 2I R − iSL1 R

ν2 = VCC − iSL2 R(3.4)

This state will be over (transistors will change state), when

νC1 = ν2 − ν1 + νB E2O N − νB E1O N

νC1 = VCC − iSL2 R − [VCC − 2I R − iSL1 R] = 2I R − R(iSL2 − iSL1) (3.5)

Since the switching occurs with a small value of R(iSL2 − iSL1), the maximumvalue of νC1 is νC1 = 2I R. Using (3.3) and (3.5), we can conclude that the amplitudeof νC1 does not change due to the coupling. The same result can be determined forνC2, 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 pro-vides a trigger signal to the other, due to the coupling, and tries to modify theamplitude and period of the other oscillator. At the end of a transient period bothrelaxation oscillators will have different amplitudes but the same frequency, differ-ent from f0. This change of amplitude and frequency due to mismatches will beanalysed 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 forthe circuit in Fig. 3.9, with mismatches, and we calculate the quadrature error. Thisanalysis is important to understand how the amplitudes change in order to preservethe quadrature relationship, which explains why this oscillator has very accuratequadrature. In the following derivation we assume that the collector resistors R areidentical in the two coupled oscillators.

We consider that the trigger in one oscillator occurs instantly when the capac-itor voltage νC in the other oscillator goes through zero. In reality, the switchingoccurs after a small delay, which we assume to be much lower than the period.This approximation is valid for small relative mismatches in the circuit components(�C/C << 1 and �I/I << 1) and for higher mismatches if we increase the cou-pling gain.

To simplify the calculations we assume that the oscillators are initially identical,without mismatches, and the capacitor voltages (νC1 and νC2) are triangular wave-forms shifted by 90◦. The waveform in advance can be either νC1 or νC2, dependingon the oscillator that has the higher stand-alone oscillation frequency. In the follow-ing analysis, we consider the general case in which C1 and C2 are different, and thecurrents I1C , I2C , I1D , and I2D are all different (Fig. 3.11).

We consider that the new values of capacitances, and charge and discharge cur-rents, occur by deviation from their ideal values. When all changes have sequen-tially been considered we redo the calculations and obtain the same amplitude at thebeginning and at the end of the period, which shows that the circuit is in steady stateoscillation. The mismatches in capacitances and in the currents are:


Fig. 3.11 Waveforms in aquadrature oscillator withmismatches

vC1

RI2−

vC2

t1t0 t2 t3 t4

T1 T3T2 T4

vC2max

vC2min

vC1min

vC1max

slopeC1

I1C

slopeC2

I2C

slopeC1

I1D

slopeC2

I2D

2R(I + ISL2)

2R(I + ISL1)

–2R(I + ISL2)

–2R(I + ISL1)

v3 − v4

v1 − v2

{

C1 = C + �C

C2 = C − �C(3.6)

⎧

⎪⎪⎪⎨

⎪⎪⎪⎩

I1C = I + �I1

I1D = I − �I1

I2C = I + �I2

I2D = I − �I2

(3.7)

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

νC2(t) = −2I R + I2C

C2t (3.8)


and from νC2(t1) = 0 we obtain:

t1 = 2RC2I

I2C(3.9)

The capacitor in the first oscillator, with capacitance C, continues to discharge af-ter t = 0, with current I. At the instant t1 the discharge finishes (the required changeof the voltage at the collectors of the first relaxation oscillator will be providedinstantly by the currents iSL1 and iSL2; this approximation is valid for a high cou-pling gain.

νC1(t1) = − I

Ct1 = −2I R

C2

C

I

I2C(3.10)

After t = t1 the capacitance of the first relaxation oscillator becomes C1. Simul-taneously the charging current becomes I1C. For t > t1:

νC1(t) = −2I RC2

C

I

I2c+ I1C

C1(t − t1) (3.11)

Since νC1(t2) = 0,

t2 = 2RC2I

I2C

[

1 + C1

C

I

I1C

]

(3.12)

and replacing (3.12) in (3.8),

νC2(t2) = 2I RC1

C

I

I1C(3.13)

At t = t2 the discharge current of the second relaxation oscillator becomes I2D ,

νC2(t) = 2I RC1

C

I

I1C− I2D

C2(t − t2) (3.14)

and, since νC2(t3) = 0

t3 = 2RC2

[I

I2C+ C1

C

(I 2

I1C I2C+ I 2

I1C I2D

)]

(3.15)

From (3.11):

νC1(t3) = −2I RC2

C

I

I2C+ I1C

C1(t3 − t1) (3.16)


and from (3.15) and (3.9):

t3 − t1 = 2RC2C1

C

I

I1C

(I

I2C+ I

I2D

)

(3.17)

Replacing (3.17) in (3.16), leads to:

νC1(t3) = 2RIC2

C

I

I2D(3.18)

We assume that when t = t3 the discharge current in the first relaxation oscillatorbecomes I1D . For t > t3:

νC1(t) = 2RIC2

C

I

I2D− I1D

C1(t − t3) (3.19)

Since νC1(t4) = 0,

t4 = 2RC2

[I

I2C+ C1

C

(I 2

I1C I2C+ I 2

I1C I2D+ I 2

I1D I2D

)]

(3.20)

From (3.14):

νC2(t4) = 2I RC1

C

I

I1C− I2D

C2(t4 − t2) (3.21)

and from (3.20) and (3.12)

t4 − t2 = 2RC2C1

C

I

I2D

(I

I1D+ I

I1C

)

(3.22)

Thus,

νC2(t4) = −2I RC1

C

I

I1D(3.23)

Now all parameters are introduced, and to find the steady-state values we mustuse 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 previoussequence of calculations we obtain:

t1 = 2R

(C1C2

C

)I 2

I1D I2C(3.24)


t2 = 2R

(C1C2

C

)(I 2

I1C I2C+ I 2

I1D I2C

)

(3.25)

t3 = 2R

(C1C2

C

)(I 2

I1C I2C+ I 2

I1D I2C+ I 2

I1C I2D

)

(3.26)

t4 = 2R

(C1C2

C

)(I

I1C+ I

I1D

)(I

I2C+ I

I2D

)

(3.27)

The oscillator is in steady state oscillation (the starting voltage and the final volt-age 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 canobtain the time intervals that are used to calculate the duty-cycle, quadrature rela-tionship, 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)

Fig. 3.12 Oscillatorsteady-state waveforms

t

vC1,2

vC2

vC1

t1 t2 t3 t4

I1D

IC

C1−2IRI2C

IC

C2−2IR

I1D

IC

C1−2IR

I1C

IC

C12IRI2D

IC

C22IR


The duty-cycles are defined as:

dc1 = T1 + T2

T= I1C I2D + I1D I2D

(I1C + I1D)(I2C + I2D)(3.33)

dc2 = T1 + T4

T= I1C I2D + I1C I2C

(I1C + I1D)(I2C + I2D)(3.34)

From the above equations we can conclude that the duty-cycle is 50% only if thecurrents that charge and discharge the capacitors are equal. The duty-cycle does notdepend on the capacitance values.

The phase difference () in a square-wave without 50% duty-cycle is not clearlydefined. We have used the definition shown in Fig. 3.13.

A phase difference applies to sinusoidal waveforms; however, the phase differ-ence of two square waveforms, can be defined as shown in Fig. 3.13.

=(

T1 + T2 + T3

2− T1 + T2

2

)2�

T

= �T1 + T3

T= �

I1C I2D + I1D I2C

(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)

= �

2

I 2 − �I1�I2

I 2= �

2

(

1 − �I1�I2

I 2

)(3.36)

Fig. 3.13 Definition ofphase difference of twosquare-waves

φ

T1 T2 T3 T4

2

T1 + T2

2

T2 + T3+

0

T1


Equation (3.36) is valid for small relative mismatches; it is still for higher mis-matches if the coupling gain is high. Equation (3.36) proves that the mismatchesin the currents have a second order effect on the quadrature relationship. With highrelative mismatches and low coupling gain other terms must be added to (3.36).

For outputs exactly in quadrature, the oscillators may have different capacitancesbut, from (3.35), the following conditions should be satisfied

I1C = I1D = I1 (3.37)and

I2C = I2D = I2 (3.38)

Only in this case we obtain perfect quadrature

= �

2(3.39)

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

f = 1

T= 1

2RI 2

C

C1C2

I1C I1D

(I1C + I1D)

I2C I2D

(I2C + I2D)(3.40)

Without mismatches f = f0, with f0 given by (3.1).From the previous derivations we can obtain the maximum and minimum values

of the voltages νC1 and νC2:

νC1min = νC1(t1) = −2RIC2

C

I

I2C(3.41)

νC1max = νC1(t3) = 2RIC2

C

I

I2D(3.42)

νC2min = νC2(t4) = −2RIC1

C

I

I1D(3.43)

νC2max = νC2(t2) = 2RIC1

C

I

I1C(3.44)

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


the oscillation amplitude to follow the second relaxation oscillator. Thus, the oscil-lator changes the amplitude and the oscillation frequency in order to preserve thequadrature 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 andI2D = I − �I in the second relaxation oscillator, the amplitude of νC2 is preservedand νC2max = |νC2min| = 2I R, while from (3.41) and (3.42):

|νC1min| = 2I RI

I + �I(3.45)

νC1max = 2I RI

I − �I(3.46)

The voltage νC1 has now a DC component with value:

VC1 = νC1max − |νC1 min|2

= 2I 2 R�I

I 2 − (�I )2≈ 2R�I

(3.47)

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

Fig. 3.14 Cross-coupledoscillator block diagram withvariable output levels in oneSchmitt-trigger

vINT2

2

G

–G –1 1

–1 –1

1

1

2

1

1

vINT1

–


3.3.4 Simulation Results

We simulated the quadrature cross-coupled oscillator at a high level using idealblocks with MATLAB, to confirm the amplitudes change predicted by the theoreti-cal analysis. As shown in the block diagram of Fig. 3.14, we change the integrationslope of the second oscillator by changing the Schmitt trigger outputs levels.

When we increase the integration slopes in the second oscillator, the amplitudeof 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 ap-pears (Fig. 3.15b). These simulations do not validate the theoretical analysis of the

−1.5

−1

−0.5

0

0.5

1

1.5

vINT1

vINT2

ΔV

ΔV

100 101 102 103 104 105 106 107 108 109 110−1.5

−1

−0.5

0

0.5

1

1.5

Time (s)

ΔV

vINT1

vINT2

ΔV

(b)

100 101 102 103 104 105 106 107 108 109 110Time (s)

(a)

Am

plitu

de (

V)

Am

plitu

de (

V)

Fig. 3.15 Effect of changing the integrator slopes in oscillator 2: (a) Increasing both slopes by10%; (b) Increasing the positive slope by 10% and decreasing the negative slope by 10%


previous section, but they confirm that the amplitudes change due to mismatches, aspredicted by the theoretical analysis.

In order to confirm the theoretical analysis at circuit level, we designed a 2.4 GHzoscillator using a 0.35 m CMOS technology (the circuit of Fig. 3.9, but with MOStransistors). To achieve that very high frequency, the circuit was designed with thefollowing parameters: R = 100 �, C1 = C2 = 420 f F, I1 = I2 = 3 mA, ISL =1 mA and VCC = 3 V. The transistor dimensions are: 200 m/0.35 m for the Mtransistors and 80 m/0.35 m for the MSL transistors.

The theoretical analysis presented is rigorous for low frequency, with triangu-lar waveforms; at high-frequency several other effects are present, which are verydifficult 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 circuitimplementation many parasitics are present; the output is approximately a sinewaveand 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 amplitudesare 2I R = 600 mV and the simulated amplitudes are about 400 mV. Although theabsolute value of the amplitude has a significant difference, due to high frequencyeffects, the relative changes show a good agreement with the theory:

(1) If we change the capacitors by 10% (small relative mismatches in capacitors,�C/C = 0.1 << 1), the amplitudes should also change 10%. The simulated

vINT2

ΔVvINT1

ΔV

Fig. 3.16 Effect of decreasing C2 by 10%


vINT1

vINT2ΔV

ΔV

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

values are VINT1 = 0.352 mV and VINT2 = 0.381 mV, which correspond to arelative change of about 8%.

(2) By changing the currents in the second oscillator by 10% (I2C = 3.3 mA andI2D = 2.7 mA), with small relative mismatches �I/I = 0.1 << 1, we expecta DC offset of about 10%. The simulated DC offset is about 49 mV, whichcorresponds to a relative change in the amplitude of about 12%.

3.4 Phase-Noise

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

3.4.1 Phase-noise in a Single Relaxation Oscillator

The noise in electronic circuits is usually described by the noise power spectraldensity S(xn), where xn is a noise variable (usually either voltage or current).

In the noise analysis of linear systems the following result is used [56, 57]. As-suming that an input x(t) produces an output y(t), and that the corresponding transfer

3.4 Phase-Noise 57

function is H (s) = Y (s)/X (s), a noise source xn with spectral density S(xn) at theinput produces an output noise yn with spectral density

S(yn) = |H (s)s= j�|2S(xn) (3.48)

Usually there is more than one noise source in a network. In this case the follow-ing result concerning random variables is used. If a noise variable y(t) is a linearcombination of different random variables x1(t) and x2(t),

y(t) = a1x1(t) + a2x2(t) (3.49)

and if x1 and x2 are independent, the relationship between the spectral densities is:

S(y) = a12S(x1) + a2

2S(x2) (3.50)

Note that if the variables are correlated (this is not considered in this book),there is an additional term in the above equation, and the derivations become rathercomplicated.

In the time domain the oscillator output (νOUT) is:

νOUT = Vom cos[�0t + �(t)] (3.51)

where Vom is the oscillator amplitude and �(t) contains the oscillator phase-noise.A carrier of amplitude Vom , modulated in phase by a sinusoidal signal of fre-

quency fm can be represented by [58]:

νOUT = Vom cos

[

(�0t) + � f pk

fmsin(�mt)

]

(3.52)

where � f pk is the peak frequency deviation. In this book we are interested in thestudy of oscillators in which the peak frequency deviation is the deviation of theoscillation frequency due to low frequency noise sources (� f0 = � f pk). As shownin textbooks on FM theory [58–60], for fm << f0 the phase modulation results infrequency components on each side of the carrier, with amplitude Vom� f0/2 fm :

νOUT ≈ Vom

[

cos(�0t) + � f0

2 fm[cos(�0 + �m)t − cos(�0 − �m)t]

]

νOUT| f = f0+ fm= Vom

� f0

2 fmcos(�0 + �m) (3.53)

Using equation (3.48) and the FM theory for low frequency noise sources withfm << f0 we obtain

S(νOUT)| f = f0+ fm=∣∣∣∣

Vom

2 fm

∣∣∣∣

2

S(� f0) (3.54)


Fig. 3.18 Noise analysis foran oscillator xn (t) Δf0 (t)

H(s)

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

L( fm) = S(νOUT)| f = f0+ fm

12 V 2

om

= 1

2 f 2m

S(� f0) (3.55)

Considering (Fig. 3.18) that a noise source xn(t) with a spectral density S(xn) inan oscillator will originate a frequency shift with spectral density S(� f0), it followsfrom equation (3.48):

S(� f0) = |H ( j�)|2S(xn) (3.56)

where H(s) relates the transforms of variables � f0 and xn .

By taking into account (3.55) and (3.56),

L( fm) = |H ( j�)|22 f 2

m

S(xn) (3.57)

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

f0 = I

2CV(3.58)

where I is the current that charges and discharges the capacitance C, and V is theSchmitt-trigger difference of threshold voltages. The noise sources in Fig. 3.19 arethe equivalent noise current in in parallel with current source I, and the equivalentnoise voltage νn in series with the Schmitt-trigger input. The noise sources modulatethe oscillation frequency, thus creating phase-noise.

The frequency shift due to in , assuming that in is approximately constant duringone period, i.e., assuming low-frequency noise, is obtained from (3.58):

� f0 = in

2CV= f0

in

I(3.59)

and using (3.57)

3.4 Phase-Noise 59

Fig. 3.19 Relaxationoscillator with noise sources

Schmitt-trigger

C

nv

V

in I

VCC

L( fm) = S(in)

2I 2

(f0

fm

)2

(3.60)

The frequency shift due to νn , using (3.58), is

� f0 = − I

2CV 2νn = − f0

Vνn (3.61)

and using (3.57)

L( fm) = S(νn)

2V 2

(f0

fm

)2

(3.62)

Equations (3.60) and (3.62) apply to low-frequency noise. However, we mustalso take into account the high frequency noise: the switching, which is inherent torelaxation oscillators, produces a mixing effect that will “fold back” high frequencynoise components, resulting in low-frequency phase-noise [35, 48, 51, 55].

For the case of high frequency current noise components, the resulting phase-noise is filtered out due to the integrating capacitor. Thus, the resulting phase-noiseis still given by (3.60). The high frequency voltage noise components will be domi-nant and produce the main contribution to the oscillator phase-noise [35, 48, 51].

Assuming that νn is white noise, it can be demonstrated [48,51] that the resultingphase-noise is

L( fm) = 2�4S(νn)

2V 2

(f0

fm

)2

(3.63)

where

� = Bc

2 f0(3.64)


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

3.4.2 Phase-noise in Quadrature Relaxation Oscillators

As indicated above by Fig. 3.8b a phase-noise reduction with strong coupling isexpected, since the switching of each oscillator is defined by a signal with a steeperslope (by increasing the coupling current we increase the slope); this ensures thatthe circuit is less sensitive to noise in the transition points that define the oscillatorfrequency.

We perform circuit level simulations of a 5 GHz oscillator (with and withoutcoupling) to demonstrate that the coupling improves the phase-noise performance.

It can be shown that the phase-noise of N coupled oscillators is reduced by afactor 1/N with respect to a single oscillator [54,61]. For two coupled oscillators (RCand LC), this improvement of 3 dB was confirmed in [34, 44, 62–64]. This result isvalid for two coupled oscillators with low coupling gain; if we increase the couplinggain, a further reduction of the oscillator phase-noise is expected in relaxation RCoscillators (in LC oscillators there is a noise increase as will be shown in Chapter 5).At circuit level the increase of coupling gain is performed by increasing the soft-limiter bias current.

We use a quadrature oscillator designed in a 0.18 m CMOS technology, for anoscillation frequency of 5 GHz. The circuit parameters are R = 100 �, (W/L) =100 m/0.18 m for M transistors, (W/L) = 100 m/0.18 m for MSL transistors,

104

105

106

107

108

–150

–140

–130

–120

–110

–100

–90

–80

–70

–60

–50

Offset Frequency (Hz)

Pha

se N

oise

(dB

c/H

z)

1) Stand alone2) ISL = 0.1 mA3) ISL = 3 mA

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

3.5 Conclusions 61

C = 300 fF, I = 3 mA, and ISL = 0.1 mA (low coupling gain) or ISL = 3 mA (highcoupling gain). The simulations are done with spectreRF considering ideal currentsources.

Figure 3.20 shows the simulated phase-noise for a stand-alone relaxation oscil-lator, and for coupled relaxation oscillators. We can observe that the phase-noiseof −113.85 dBc/Hz @ 10 MHz is improved by 3 dB with weak coupling. Increas-ing ISL from 0.1 mA to 3 mA gives a further reduction of the phase-noise of about6.2 dB, bringing the total reduction to 9.2 dB.

In Chapter 7 we will present experimental results for this oscillator, and we willcompare it with an LC cross-coupled oscillator designed for the same frequency andusing the same technology.

3.5 Conclusions

In this chapter we presented a detailed study of the quadrature relaxation oscillatorconcerning its key aspects: oscillation frequency, output amplitude, and quadraturerelationship. The phase-noise is investigated by simulation.

An oscillator consisting of two relaxation oscillators that are cross-coupled us-ing two coupling blocks has outputs in very accurate quadrature. The variation ofparameters in one oscillator is compensated by a variation of the amplitude of theother. Moreover, we show that a difference between the charge and discharge cur-rents in one relaxation oscillator results in a DC component in its capacitor voltage.Thus, the quadrature is preserved, and the effect of mismatches is automaticallycompensated.

We identify the noise sources and derive an equation for the phase-noise of a sin-gle relaxation oscillator at a high level. To investigate the phase-noise of quadraturecross-coupled relaxation oscillators we simulate a 5 GHz circuit, which confirmsthat there is a reduction of the oscillator phase-noise due to coupling.

From our study we can conclude that the coupling block has a strong influenceon the quadrature oscillator performance. Increasing the gain of the coupling blockwill improve the oscillator performance: the effect of mismatches on the quadratureerror is reduced, and becomes a second order effect, as shown by (3.36); the oscil-lator phase-noise is also reduced. Thus, to improve the performance of a quadraturerelaxation oscillator, the value of the coupling gain should be increased, but this hasa cost in terms of power consumption.

Chapter 4Quadrature Oscillator-Mixer

Contents

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.2 High Level Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2.1 Ideal Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.2.2 Effect of Mismatches and Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.3 Circuit Level Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.1 Introduction

In this chapter we study the possibility of including the mixing function in theoscillator, thus avoiding the use of external mixers. Conventional mixers have anopen-loop structure, in which the output is obtained by multiplication of an LOsignal with an input signal (RF signal, in the receiver path). For quadrature modula-tion/demodulation two independent mixers are necessary, and accurate matching ofcircuit elements is required to avoid quadrature errors. In the previous chapter, weproved that the coupling reduces the effect of mismatches, providing very accuratequadrature of the outputs. In this chapter we show that accurate quadrature is main-tained when the mixing function is integrated in the oscillator, without extra cost,either in area or in power consumption.

In [15, 65] a theoretical study of the quadrature relaxation oscillator was pre-sented using Matlab and with low frequency validation, and in [66] the oscillator-mixer concept was presented and demonstrated using Matlab. In this chapter wedemonstrate the same concept with a high frequency circuit implementation, and westudy the influence of the mixing function on the quadrature oscillator performance.This can be seen as an extension of the results presented in [15, 65].

In this chapter we present a high level theoretical study of the influence ofmismatches and delays on the duty-cycle, oscillation frequency, and quadraturerelationship in the oscillator-mixer. This study is done considering the couplingblock as a linear amplifier, because at the switching points the coupling block isin its linear region. For validation at circuit level we designed a 2.4 GHz quadrature


63

64 4 Quadrature Oscillator-Mixer

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

4.2 High Level Study

4.2.1 Ideal Performance

The quadrature relaxation oscillator has accurate quadrature outputs due to feed-back, and the components mismatch has only a second order effect on the frequencyand phase accuracy [15]. The goal is to obtain the mixing function in this circuitwithout disturbing the good oscillator performance. We can inject the modulatingsignal (multiplying the input signal with the oscillator signal) in three differentblocks in the model of Fig. 4.1 [66]:

1 – Integrator – If we multiply the input signal by the signal either before or afterthe integrator, the zero crossings of the oscillator output will change, which willaffect its phase and frequency stability.

2 – Schmitt-trigger – If we multiply the input signal with the signal before theSchmitt-trigger or inside of the Schmitt-trigger block (Fig. 3.2), the Schmitt-trigger decision levels will be affected, and, therefore, the integrator signals willalso be affected. As in the previous case, this will change the frequency and phaseof the oscillator.

3 – Soft-limiter – Multiplying the input signal with the signal either before or afterthe soft-limiter is the only possibility to avoid deteriorating the oscillator per-formance, because this does not change the timing of the zero-crossings of thesoft-limiters outputs, which are responsible for the quadrature relationship andfrequency of the oscillator (Fig. 4.1). The soft-limiter output provides a sort ofinjection locking, by triggering the other oscillator so that it changes at the right

Fig. 4.1 Quadraturerelaxation oscillator-mixer inan upconversionconfiguration

1 f v

2 f v

OUTv

Integrator Schmitt-trigger1

1 –1

1

–1

–1

1 –1

+

+

+

+

+

–

Soft-limiter

A

A

B

B

4.2 High Level Study 65

time. Changing the gain of the soft-limiter, or changing its saturation levels, willchange only the strength of this triggering, and the performance of the oscillatorwill not be affected as a first order effect. Thus, this circuit can be used as modu-lator and demodulator for quadrature signals.

In the high-level model of Fig. 4.1 we have basically two possibilities: to add anew input to the soft-limiter block, to control its gain, or to change the soft-limitersaturation levels. The injected signal must be associated to a DC component, toavoid a soft-limiter output change of sign.

In the block diagram in Fig. 4.1, two voltage inputs, are added, one with fre-quency f1, in soft-limiter A (I path) and the other with frequency f2 in the soft-limiter B (Q path). In the Matlab block diagram we basically add an ideal multiplier(linear modulation) after the soft-limiter in order to perform the mixing function.We have

{

ν f 1 = A + a cos(�1t)

ν f 2 = B + b sin(�2t)(4.1)

where |a| < A and |b| < B. The soft-limiter is implemented by a differential pair, inwhich a change of gain is obtained by changing the bias current: we do not needto add any DC value (the DC component is the soft-limiter bias current, whichphysically cannot be negative).

The oscillator output is an approximately square waveform with fundamental fre-quency �0 and with several harmonics at n�0 (n integer). In the following derivationwe will consider only the fundamental frequency �0 (the higher harmonics will befiltered out after the mixing). The output of the oscillator-mixer is

νOUT = An cos(�0t) cos(�1t) + An sin(�0t) sin(�2t)

= 1

2An[cos(�0 + �1)t + cos(�0 − �1)t]

+1

2An[sin(�0 + �2)t + sin(�0 − �2)t] (4.2)

The oscillator-mixer performing upconversion is simulated in Matlab, consider-ing two different modulating signals, normalized to the carrier before mixing. Theoutput is in accordance with (4.2) and is represented in the frequency domain inFig. 4.2. In a complete transmitter this signal is filtered (removing undesired higherfrequencies), amplified by the power amplifier (PA) and transmitted by the antenna.

To recover the information we need to perform a downconversion, by using theoscillator-mixer as shown in Fig. 4.3, where the RF input signal is received by theantenna and amplified by a low-noise amplifier (LNA). The RF signal (νIN) is ap-plied directly to the soft-limiters as in the upconversion case.


0.6 0.7 0.8 0.9 1 1.1 1.2 1.3–100

–90

–80

–70

–60

–50

–40

–30

–20

–10

0

10

Frequency (Normalized to Carrier Frequency)

Am

plitu

de (

dB)

0f

f0 − f2 f0 + f2

f0 − f1 f0 + f1

Fig. 4.2 IQ modulation with I at f1 and Q at f2 (Matlab simulation)

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

νI = 1

4An cos(�1) + terms with higher frequencies (4.3)

νQ = 1

4An sin(�2) + terms with higher frequencies (4.4)

Fig. 4.3 IQ oscillator-mixerin a downconversionconfiguration

INv

Iv

Qv


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5–110

–100

–90

–80

–70

–60

–50

–40

–30

–20

–10

0

Frequency (Normalized to the Carrier)

Am

plitu

de (

dB)

I SignalQ Signal

1f 2f

Fig. 4.4 IQ demodulation with I at f1 and Q at f2

To recover the I ( f1) and Q ( f2) signals we only need to apply low-pass filteringto the soft-limiter outputs to remove the high frequency terms in (4.3) and (4.4).To perform correctly the demodulation, avoiding cross-talk between the channels,the demodulator’s oscillation frequency and phase must be the same as those in themodulator. To synchronize the two oscillators, in the modulator and demodulator,a PLL should be used. Traditional PLLs (using a mixer and a low-pass filter) onlysynchronize the frequency and not the phase, while a charge-pump PLL can syn-chronize both frequency and phase [1].

Simulation of the complete system with modulation and demodulation, using aPLL to synchronize the oscillators, produces only the f1 signal in the I path andonly the f2 signal in the Q path, as expected; this is shown in Fig. 4.4.

4.2.2 Effect of Mismatches and Delay

In the previous section we have considered that the two coupled oscillators are iden-tical. In this section we will study the oscillator-mixer quadrature relationship andoscillation frequency with mismatches (the amplitude will be considered only in thestudy at circuit level). In a practical implementation, mismatches between compo-nents and other disturbances, such as delays, produce variation of the frequency andphase difference.

To investigate the effect of including the mixing in the oscillator, we considertwo signals with different frequencies at the input of the soft-limiters, x1(t) and


Fig. 4.5 Quadraturemodulation with I signal atfrequency f1 and Q signal atfrequency f2

iAX

lAD

lBD

hAD

hBD

iBX

B

aAX

aBX

lAX

lBX

1

(t)2

oAX

oBX

τB

τA

+ –1

x

(t)x

–

x2(t). We assume that the soft-limiters are in their linear region, so any change ofthe saturation levels is not relevant: what matters is the gain in the linear region.

In the following analysis we consider that the soft-limiter acts as a linear am-plifier and that the modulating signal changes its gain. The block diagram, withthe variables marked, is represented in Fig. 4.5, and the variables are defined inTable 4.1.

In the following derivation we consider that the two input frequencies aref1 << f0 and f2 << f0, i. e., we assume that the two input signals are approx-imately constant in one period, which is only valid for the upconversion case(Fig. 4.1). In the case of downconversion (Fig. 4.3) the analysis is much morecomplicated, because the input frequency is of the same order of the oscillationfrequency (this is not studied in the book).

In Fig. 4.6 we represent the integrator and Schmitt-trigger waveforms, in quadra-ture (note that the waveforms are symmetric, which is only valid for equal oscillatorsand without delays), and we mark the time instants and time intervals used in theanalysis. Exact equations for the time intervals T1, T2, T3, and T4 as a function of

Table 4.1 Circuit parameters

Xi{A,B} Output of integrator A or BXs{A,B} Output of Schmitt-trigger A or BXl{A,B} Output of soft-limiter A or BXa{A,B} Output of adder A or B

K {+,−}i{A,B} Constant of integrator A or B (positive or negative slopes)

Gl{A,B} Gain of soft-limiter A or BDl{A,B} Low decision level of Schmitt-trigger A or BDh{A,B} High decision level of Schmitt-trigger A or Bτ{A,B} Delay in cross-coupling path A or Bx1, x2 Input signals


iAX

iBX

2T 3T4T1T 1T

0t 1t 2t 3t 4t

XoAX

oBX

1

1−

1

1−

t

t

t

o{A,B}

Xi{A,B}

Fig. 4.6 Oscillator waveforms

the circuit parameters are determined using an approach similar to that in [15, 65],where the oscillator was studied without mixing.

In order to calculate the four time intervals shown in Fig. 4.6 we need to deter-minate a system of equations for each of the four states. The equations are obtainedusing a symbolic algebraic program (Derive�).

Determination of T1

We assume that in the state that lasts T1 in Fig. 4.6 the Schmitt-trigger outputsare XoA = 1 and XoB = 1, and therefore, the integrator inputs are positive. Theintegrator outputs, at t = t1, considering the effect of delay in the cross-couplingpaths of the oscillator system are:

Xi A(t1) = Xi A(t0) + K +i AT1 (4.5)

Xi B(t1 − τb) = Xi B(t0) + K +i B(T1 − τb) (4.6)


The soft-limiter B output, responsible for the triggering of oscillator A is modu-lated by input signal x2. In the following analysis we consider that the soft-limitergains are not constant: they are modulated by two input signals x1(t) and x2(t) witha frequency much lower than the oscillation frequency. The soft-limiter output valueat t1 is:

Xl B(t1) = Gl B x2(t)Xi B(t1 − τB) (4.7)

The output of the adder A for t = t1 is:

Xa A(t1) = +Xi A(t1) + Xl B(t1) (4.8)

This is the input to the Schmitt-trigger A. The oscillator will change state (theSchmitt-trigger output will change) when Xa A reaches the higher threshold level:

Xa A(t1) = Dh A (4.9)

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

T1 = Dh A − Gl B x2(t)Xi B(t0) − Xi A(t0) + Gl B x2(t)K +i BτB

Gl B x2(t)K +i B + K +

i A

(4.10)

Determination of T2

In the next time interval (T2) the Schmitt-trigger outputs have the values XoA = −1and XoB = 1. Thus,

Xi A(t2 − τA) = Xi A(t1) − K −i A(T2 − τA) (4.11)

Xi B(t2) = Xi B(t0) + K +i B(T1 + T2) (4.12)

The output of soft-limiter A, responsible for the triggering of oscillator B is mod-ulated by signal x1. The soft-limiter output at t = t2 is:

Xl A(t2) = Gl Ax1(t)Xi A(t2 − τA) (4.13)

The output of adder B for t = t2 is:

XaB(t2) = Xi B(t2) − Xl A(t2) (4.14)

This is the input to the Schmitt-trigger block B. The oscillator will change statewhen XaB reaches the higher threshold level:

XaB(t2) = Dh B (4.15)


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

T2 = Dh B + Gl Ax1(t)Xi A(t1) − K +i B T1 − Xi B(t0) + Gl Ax1(t)K −

i AτA

Gl Ax1(t)K −i A + K +

i B

(4.16)

Determination of T3

In the third time interval (T3) the Schmitt-trigger outputs have the values XoA = −1and XoB = −1. Thus,

Xi B(t3 − τB) = Xi B(t2) − K −i B(T3 − τB) (4.17)

Xi A(t3) = Xi A(t1) − K −i A(T2 + T3) (4.18)

The output of the soft-limiter B is modulated by the input signal x2. The soft-limiter output at t = t3 is:

Xl B(t3) = Gl B x2(t)Xi B(t3 − τB) (4.19)

The output of the adder A for t = t3 is:

Xa A(t3) = Xi A(t3) + Xl B(t3) (4.20)

This is the input to the Schmitt-trigger A. The oscillator will change state whenXa A reaches the lower threshold level:

Xa A(t3) = Dl A (4.21)

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

T3 = −Dl A + Gl B Xi B(t2)x2(t) − K −i AT2 + Xi A(t1) + Gl B x2(t)K −

i BτB

Gl B K −i B x2(t) + K −

i A

(4.22)

Determination of T4

In the fourth and last time interval (T4) the Schmitt-trigger outputs are XoA = 1 andXoB = −1. Thus,

Xi A(t4 − τA) = Xi A(t3) + K +i A(T4 − τA) (4.23)

Xi B(t4) = Xi B(t2) − K −i B(T3 + T4) (4.24)


The soft-limiter A output is modulated by the first input signal x1. The soft-limiteroutput at t = t4 is:

Xl A(t4) = Gl Ax1(t)Xi A(t4 − τA) (4.25)

The output of the adder B for t = t4 is:

XaB(t4) = Xi B(t4) − Xl A(t4) (4.26)

This is the input to the Schmitt-trigger B. The oscillator will change state whenXaB reaches the lower threshold level:

XaB(t4) = Dl B (4.27)

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

T4 = −Dl B − Gl Ax1(t)Xi A(t3) − K −i B T3 + Xi B(t2) + Gl Ax1(t)K +

i AτA

Gl Ax1(t)K +i A + K −

i B

(4.28)

Oscillation frequency and phase error

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

K +i A(T1 + T4) = K −

i A(T2 + T3) (4.29)

K +i B(T1 + T2) = K −

i B(T3 + T4) (4.30)

With equations (4.29–4.30) we can obtain T1, T2, T3, and T4. These can be usedto calculate the duty-cycle, oscillation frequency, and phase difference relationshipof the outputs.

We use the following definitions:(1) Duty-cycle

dcA = T1 + T4

Tand dcB = T1 + T2

T(4.31)

(2) Oscillation frequency

f0 = 1

T1 + T2 + T3 + T4(4.32)


(3) Phase difference

= 180T1 + T3

T(4.33)

We also use the following notation

Ki A = K +i A + K −

i A

2Ki B = K +

i B + K −i B

2(4.34)

�Ki A = K +i A − K −

i A

2�Ki B = K +

i B − K −i B

2(4.35)

DA = Dh A − Dl A DB = Dh B − Dl B (4.36)

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

dcA = 1

2

(

1 − �Ki A

Ki A

)

dcB = 1

2

(

1 − �Ki B

Ki B

)

(4.37)

= 90◦

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 − �Ki A

Ki A

�Ki B

Ki B

+[

DB

Gl Ax1(t)DA+ 2Ki A

DAτA

]

[

1 −(

�Ki A

Ki A

)2]

1 + Gl B x2(t)Ki B DB

Gl Ax1(t)Ki A DA+ 2Gl B x2(t)Ki B

DA(τA + τB)

−[

DA

Gl B x2(t)DB+ 2Ki B

DBτB

]

[

1 −(

�Ki B

Ki B

)2]

1 + Gl A Ki A DAx1(t)

Gl B Ki B DB x2(t)+ 2Gl Ax1(t)Ki A

DB(τA + τB)

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(4.38)

f0 = 1

2

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Ki A

DA

[

1 −(

�Ki A

Ki A

)2]

1 + Gl B x2(t)Ki B DB

Gl Ax1(t)Ki A DA+ 2Gl B x2(t)Ki B

DA(τA + τB)

+ Ki B

DB

[

1 −(

�Ki B

Ki B

)2]

1 + Gl Ax1(t)Ki A DA

Gl B x2(t)Ki B DB+ 2Gl Ax1(t)Ki A

DB(τA + τB)

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(4.39)


From (4.37–4.39) we arrive at the following conclusions for the oscillator-mixer:(1) The integration constant, Ki , controls the oscillator duty-cycle. The intro-

duction of the mixing function does not affect the duty-cycle.(2) If we consider only mismatches in the integrator difference of positive and

negative slopes �Ki A and �Ki B , and if the coupling gain is very high and withx1(t) = x2(t) = 1, equation (4.38) simplifies to

= 90◦[

1 − �Ki A

Ki A

�Ki B

Ki B

]

(4.40)

which means that the quadrature error is the product of two relative errors, becominga second order error term.

(3) In order to make it easier to see the meaning of (4.38) we consider that thesame modulating signal is injected in both soft-limiters x(t) = x1(t) = x2(t), andwe make the following simplifying assumptions:

(3a) If the only non-ideal effect is a mismatch between the two integration con-stants, Ki A and Ki B ,

= 90◦[

1 + 1

Gl x(t)

Ki A − Ki B

Ki A + Ki B

]

(4.41)

(3b) If the only non-ideal effect is the mismatch of the difference between thedecision values of the Schmitt-triggers, DA and DB,

= 90◦[

1 + 1

Gl x(t)

DB − DA

DA + DB

]

(4.42)

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

= 90◦[

1 − �A − �BDKi

+ Gl x(t)(�A + �B)

]

(4.43)

Equations (4.41–4.43) show that mismatches and delays in the circuit, which areresponsible for a first order effect on the quadrature error, are attenuated by thesoft-limiter gain. For example if we have a 10% deviation in one parameter, withx1(t) = x2(t) = 1, and the soft-limiter gain is 10, the quadrature error is about 1◦,which is a low value.

Equations (4.41–4.43) also show that the modulation signal has some influenceon the phase difference. If there are mismatches and modulating signal some phase-noise is originated; by increasing the coupling gain this influence can be neglected.

(4) Without any mismatches equation (4.39) can be simplified to

f0 = Ki

2D + 4Gl x(t)Ki �(4.44)

4.3 Circuit Level Study 75

If the delay is neglected (� = 0), f0 = Ki2D , which is the ideal result obtained in

Chapter 3.(5) The soft-limiter gain is a critical parameter. It should be high to minimize

variations in frequency and phase. However, a high Gl reduces the maximum os-cillation frequency in the presence of delays, as shown by equation (4.44). Sincedelays are usually small, the reduction of frequency can be neglected.

4.3 Circuit Level Study

In order to validate the previous high level analysis at circuit level we designed a2.4 GHz oscillator-mixer with the circuit in Fig. 4.7. In this section we will not do atheoretical study at circuit level, but, instead we will perform simulations concerningthe following parameters: quadrature relationship, gain, oscillation frequency, andphase-noise. Other mixer parameters, such as intermodulation distortion (IIP2 andIIP3), 1-dB compression point, input and output impedance, noise figure, and LOleakage, need further analysis.

To achieve an oscillation frequency of 2.4 GHz the circuit was designed with thefollowing parameters: R = 100 �, M transistors with (W/L) = 200 m/0.35 m,and soft-limiter transistors MSL , with 80 m/0.35 m, C1 = C2 = 420 fF, I1 =I2 = 3 mA, and ISL = 1 mA, supply voltage VCC = 3 V (this is the same circuit ofthe example in Chapter 3).

The results in Figs. 4.8 and 4.9 are obtained with iSL = 1 + 0.2 cos(�1t) [mA]with f1 = 300 MHz. In Figs. 4.8 and 4.9 we represent the oscillator outputs (νOUT1

M

R R

1I SLi

R R

2C

2I SLi

M M M

MSLMSLMSL MSL

2I

1C

1I

CCV

OUT1v OUT2v

Fig. 4.7 Oscillator-mixer circuit implementation


0 1 2 3 4 5 6−90

−80

−70

−60

−50

−40

−30

−20

−10

0

Frequency (GHz)

)V

Bd(edutilp

mA

f0

f0 – f1 f0 + f1

Fig. 4.8 Oscillator-mixer output in the frequency domain

10 11 12

0.8

13 14 15 16 17 18 19 20−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

OUT2vOUT1v0.6

Time (ns)

)V(

edutilpm

A

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

4.3 Circuit Level Study 77

and νOUT2), in the frequency and in the time domains, which show that this cross-coupled oscillator, can be used as a combined oscillator-mixer. At circuit level, whenwe change the limiter bias current we change its gain and the limiting levels at thesame time. The dominant effect is the changing of the limiting levels (the change inthe coupling gain can be neglected).

An amplitude modulated (AM) carrier with modulating signal x(t) is [58]:

xc(t) = Ac[1 + x(t)] cos(�0t) (4.45)

where Ac is the unmodulated carrier amplitude, and is a positive constant (modu-lation index).

In the circuit of Fig. 4.7 the amplitude of the output square-waves is

Vom = 4R(I + iSL ) (4.46)

Since I = 3 mA, and iSL = 1 + 0.2 cos(�1t), the modulation index is = 0.05.In Figs. 4.9 and 4.10 (Fig. 4.10 is a zoom of Fig. 4.9) we can observe the two

oscillator outputs in the time domain. The mixer gain is about 500 mV/mA (theinput is a sinusoidal current with 0.2 mA amplitude and the output is a sinusoidalvoltage with 100 mV amplitude). The modulation index is about 0.1, which is thedouble of the expected value. This difference can be explained by noting that, athigh frequency, the waveforms are approximately sinusoidal with lower amplitudethan in the theory (about half in our simulations).

20 20.1 20.2 20.3 20.4 20.5 20.6 20.7 20.8 20.9 21−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Time (ns)

)V(

edutil pm

A

OUT1v OUT2v

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


Table 4.2 Effect of 10% increase in current I1 (Fig. 4.7)

ISL (mA) Oscillator Oscillator-mixer ( = 0.05)

Frequency (GHz) Phase Difference (◦) Frequency (GHz) Phase Difference (◦)

1 2.42 91.9 2.42 92.010 2.14 90.2 2.14 90.3

Table 4.3 Effect of 10% decrease in current I1 (Fig. 4.7)



1 2.38 88.2 2.38 88.110 2.14 89.8 2.14 89.7

Table 4.4 Effect of 10% increase in C1 (Fig. 4.7)



1 2.35 85.7 2.35 85.410 2.10 89.2 2.10 88.7

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



1 2.44 93.8 2.44 94.010 2.19 90.7 2.19 90.8

The influence of mismatches on the oscillator-mixer frequency and phase rela-tionship is shown by the simulation results in Tables 4.2–4.5. In order to change thesoft-limiter gain we change the bias current of the differential pair.

In Tables 4.2 and 4.3 we observe the influence on the quadrature relationshipof a change in the currents in one oscillator. We consider the oscillator withoutmodulation and with modulation index 0.05 for the two cases: ISL = 1 mA andISL = 10 mA. In Table 4.2 we consider a 10% increase in the currents in one oscil-lator (I1 = 3.3 mA) when the currents in the other oscillator have the nominal value(I2 = 3 mA). In Table 4.3 we consider a 10% decrease of I1 (I1 = 2.7 mA).

We also consider the influence of mismatches in the capacitors (which are alsoresponsible for the integration constant). In Table 4.4 we consider a 10% increasein C1(C1 = 462 fF), and in Table 4.5 we consider a 10% decrease (C1 = 378 fF),while C2 remains with the nominal value (C2 = 420 fF).

From Tables 4.2–4.5 we observe that the increase in the limiter gain (by increas-ing ISL from 1 mA to 10 mA) reduces the oscillation frequency, which is expected

4.4 Conclusions 79

105 108107106–150

–140

–130

–120

–110

–100

–90

–80


Pha

se N

oise

(dB

c/H

z)

OscillatorOscillator-Mixer

Fig. 4.11 Oscillator phase-noise with and without mixing

from (4.44), and improves the quadrature accuracy, both in the oscillator and in theoscillator-mixer, in accordance with (4.38) and (4.41).

In Fig. 4.11 we plot the oscillator phase-noise with and without modulation. Weobserve that the mixing leads to some degradation of the oscillator phase-noise, aspointed out above; however, this degradation is negligible (in our simulation thedegradation is about 1 dB).

4.4 Conclusions

In this chapter we show that a quadrature relaxation oscillator can perform the mix-ing function if we inject a modulating signal in the oscillator feedback loop, bychanging the soft-limiter gain and/or limitation levels.

In a high level study, we derive equations for the duty-cycle, quadrature rela-tionship, and oscillation frequency of the quadrature oscillator-mixer. We show thatthe effect of mismatches is attenuated, and becomes a second order effect. This isthe main advantage of combining the oscillator and mixer, and is in contrast with theuse of separate mixers, in which case mismatches are responsible for a first orderquadrature error. The approach proposed here has also the important advantage ofreducing the area and power consumption.

A 2.4 GHz CMOS relaxation oscillator-mixer was designed with a 0.35 mCMOS AMS technology to verify the theoretical study presented in [15, 65] for


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

Chapter 5Quadrature LC-Oscillator

Contents

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.2 Single LC Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.3 Quadrature LC Oscillator Without Mismatches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.4 Quadrature LC Oscillator with Mismatches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.5 Q and Phase-Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.6 Quadrature LC Oscillator-Mixer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.1 Introduction

Quadrature oscillator structures with feedback have been shown to provide accu-rate quadrature outputs. In Chapter 3 we studied the quadrature relaxation oscillatorcomposed of two cross-coupled RC oscillators. In this chapter we study the quadra-ture LC oscillator composed of two cross-coupled LC oscillators.

In a relaxation oscillator the transistors act as switches, and the operation ishighly nonlinear. These oscillators are known to have higher phase-noise than LCoscillators, which operate approximately linearly. Recently [11] the coupling of twoLC oscillators was proposed to obtain quadrature outputs, and the study of thistype of oscillators is still in its initial stage. In this chapter we study the effect ofcross coupling two LC oscillators, and we evaluate the effect of mismatches onthe oscillation frequency (which can be different from the free-running frequencyof the individual oscillators); we also study the effect on Q and the degradationof phase-noise. This phenomenon, phase-noise degradation due to coupling, wasobserved independently by some of the authors of the present book and by anotherresearch group; these findings have been reported simultaneously [20, 63].

Single LC oscillators have better phase-noise performance than single relaxationoscillators; however, when LC oscillators are coupled to provide quadrature outputs,there is some phase-noise degradation. We show that a strongly coupled relaxationoscillator can have a phase-noise performance similar to that of strongly coupledLC oscillators.


81

82 5 Quadrature LC-Oscillator

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

5.2 Single LC Oscillator

The basic circuit of an LC tuned oscillator and the basic model to predict phase-noise were summarized in Chapter 2. In this section we consider the noise sourcesand calculate the phase-noise of a single LC oscillator. This oscillator (representedin Fig. 5.1) has been of interest for a long time [67–74], and, due to its low phase-noise, it was extensively studied [75–82].

A simple equation for the tank voltage amplitude can be determined assumingthat the differential pair in steady-state switches the tail current (IT ) into eitherbranch of the LC resonator [43]. The differential pair is modelled as a current sourcewith a current that jumps between two extreme values (assuming that the transistorsare ideal switches), connected in parallel with the RLC tank. It is assumed that Rp

is the equivalent parallel tank resistance (Fig. 5.2).Considering the model of Fig. 5.2, the voltage harmonics are strongly attenuated

by the LC tank, and at the resonance frequency the impedances of inductor andcapacitor cancel, leaving only the parallel resistance. The LC tank acts as a filter[43]: the square-wave current, represented in Fig. 5.3 a, generates an approximatelysinusoidal voltage across the resonator, shown in Fig. 5.3b, with amplitude:

Fig. 5.1 LC oscillator

VCC

L 2

M M

L 2

IT

2C 2C

5.2 Single LC Oscillator 83

Fig. 5.2 Equivalent circuit ofthe LC oscillator

i(t) v(t)CLRp

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

t

i(t)

t

v(t)

Vtank

–Vtank–IT

IT

a) b)

Vtank ≈ 4

�IT Rp (5.1)

For the circuit in Fig. 5.1, if the oscillation is at very high frequency and theoscillator has low voltage amplitude, a better approximation is to assume that thedifferential pair is working in the linear region. In this mode of operation, referredto in the literature as current-limited, the tank current is approximately sinusoidaland the voltage amplitude is lower than (5.1), about IT Rp [83].

In an LC oscillator the noise is originated in three different blocks: the lossy LCtank, the transistors of the differential pair, and the tail current source.

We can calculate the phase-noise contribution of the tank assuming that the onlynoise source is the thermal noise [2], which is represented either as a current sourceacross the tank with a spectral density,

S(in) = 4kT

Rp(5.2)

or as a voltage noise source in series with the tank with spectral density

S(νn) = S(in)|Z |2 (5.3)

where Z is the tank impedance.Using the model of Fig. 5.2, for small offset frequencies with respect to the fun-

damental frequency (�m � �0/2Q), it can be shown that the impedance of a LCtank can be approximated by [1, 2]

|Z (�0 + �m)|2 ≈ R2p

1

4Q2

(�0

�m

)2

(5.4)


We use the definition Q [36]

Q = �0

2

√(

d A

d�

)2

+(

d�

d�

)2

(5.5)

where A = |Z ( j�)|, � = arg |Z ( j�)|, and �0 = 1/√

LC is the resonance fre-quency.

In an LC oscillator d A/d� = 0 [36], and for Q0 = Q(�0)

Q0 = �0

2

∣∣∣∣

d�

d�

∣∣∣∣

∣∣∣∣�=�0

= Rp

√

C

L= Rp

�0L(5.6)

Considering that the losses in the capacitors are much lower than those in theinductors, the resonator quality factor is determined mainly by the inductor, and theparallel resistance is obtained from the inductor quality factor [2].

Using (5.3) and (5.4),

S(νn) = 4kT

Rp

∣∣∣∣Rp

�0

2Q�m

∣∣∣∣

2

= 4kT Rp

(�0

2Q�m

)2

(5.7)

From (5.7) we can conclude that increasing Q leads to a reduction in the noisespectral density, when all the other parameters remain unchanged. The output noiseis frequency dependent, due to the filtering action of the tank: the spectral density isinversely proportional to the square of the offset frequency. This behavior is due tothe fact that the voltage frequency response of an RLC tank rolls off as 1/f to eachside of the center frequency, and power is proportional to the square of voltage [67].

An important aspect is that thermal noise affects both amplitude and phase, andequation (5.7) includes their combined effect. The equipartition theorem of thermo-dynamics [84] states that in equilibrium, and in the absence of amplitude limiting,the amplitude-noise and phase-noise powers are equal (the noise energy is splitequally into amplitude and phase) [67]. We know that oscillators have an amplitudelimiting mechanism and this will remove most of the amplitude-noise. Therefore,the total noise power in the oscillator will be approximately half the noise givenby (5.7) [1, 2, 34].

The phase-noise spectral density is usually divided by the carrier power,

L(��) = 10 log

[

2kT

Pcarrier

(�0

2Q�m

)2]

(5.8)

where Pcarrier = V 2tank/Rp is the carrier power. It should be noted that (5.8) is only

valid for the 1/ f 2 region of the noise spectrum (Fig. 2.15). A complete equation forall the spectral oscillator regions was presented by Leeson [39].

5.3 Quadrature LC Oscillator Without Mismatches 85

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

L(��) = 10 log

[

2k FT

Pcarrier

(�0

2Q�m

)2]

(5.9)

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

5.3 Quadrature LC Oscillator Without Mismatches

The simplest and most used implementation of the LC oscillator uses transistorsto generate the negative conductance as represented in Fig. 5.1. However, severalvariations of this implementation in order to improve the oscillator phase-noise per-formance, can be found in the literature [85–92]

Lately, there has been some research in coupled LC oscillators and several ap-proaches have been presented [93–97]. The implementation in Fig. 5.4, first pre-sented in [11], couples two equal LC oscillators, expecting to inherit the good phase-noise performance of the individual oscillators. The coupling block is implemented,as in the relaxation oscillator, with a differential pair that senses the voltage at oneoscillator output and injects a current in the second oscillator, in order to trigger it.

A linear model of a cross-coupled LC oscillator consists of two coupled parallelRLC circuits as represented in Fig. 5.5. Without mismatches, C1 = C2 = C , L1 =L2 = L , and Rp1 = Rp2 = Rp. In parallel with each tank there are negativeresistances −1/gm , which cancel the losses. Two differential transconductances gmc

provide the coupling, and are responsible for the quadrature outputs.In the linear model of Fig. 5.5 the loop gain is:

G loop(s) = −g2mc

⎛

⎜⎜⎝

sL

1 + sL

(1

Rp− gm

)

+ s2LC

⎞

⎟⎟⎠

2

(5.10)

Using the Barkhausen criterion for the loop gain we equate (5.10) to 1 with gm =1/Rp, and we solve in order to � (s = j�).

From

± j = gmcsL

1 + s2LC(5.11)


2L

M M

2L

IT

VCC

M M

IT

MSL

ISL1

MSL MSL

ISL2

MSL

v1

2L2L

v2

2C2C C2 C2

Fig. 5.4 Quadrature LC Oscillator

C2 Rp2

gmc

L2

gmc

i2

v2gm

1

C1 Rp1L1

i1

v1gm

1

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

5.3 Quadrature LC Oscillator Without Mismatches 87

making s = j�

±1 = gmc�L

1 − �2LC(5.12)

�2 ± gmc

2C2�L − �2

0 = 0 (5.13)

From (5.13) the oscillation frequency has two solutions:

�osc1 = +gmc

2C+ �0

√(

1 + g 2mc L

4C

)

(5.14)

�osc2 = −gmc

2C+ �0

√(

1 + g 2mc L

4C

)

(5.15)

Assuming that,

g 2mc L

4C� 1 (5.16)

(this is always satisfied with practical values) we obtain,

�osc1 ≈ �0 + gmc

2C(5.17)

�osc2 ≈ �0 − gmc

2C(5.18)

From equations (5.17) and (5.18) we can observe that coupling two oscillatorswill produce some shift in the oscillation frequency, which is in accordance withprevious publications [63, 69, 93]. To find out which of the two solutions, (5.17)or (5.18), will prevail, a study of the stability of the oscillations is required, which,to our knowledge, has not been done so far (that is an interesting topic for futureresearch work). In the measurements (to be presented in Chapter 7) we observe thatit is the lower frequency that prevails.

Using the Barkhausen criterion, the phase of the loop gain is a multiple of 2�.This means that the argument of the impedance of the resonant circuit is = +�/2or = −�/2, as required for quadrature outputs.

The impedance of a parallel RLC circuit with the compensating conductance−1/gm is:


Z (s) = sL|| 1

sC||Rp|| − 1

gm

= 11

j�L+ j�C + 1

Rp− gm

= j�L

1 − �2LC + j�L

(1

Rp− gm

) (5.19)

and the argument is:

� = arg[Z ] = �

2− tan−1

�L

(1

Rp− gm

)

(1 − �2LC)(5.20)

Fig. 5.6 Tank impedancephase with and withoutcompensation of losses

ω0 ω

θ

LC

RLC2π

2

π−

Fig. 5.7 Phasor diagramwithout mismatches

V2gmc

V1gm

I1

θ

φ

θ V2gm

V1gmc−

I2

V1

V2

5.4 Quadrature LC Oscillator with Mismatches 89

In Fig. 5.6 we plot the two extreme cases, without compensation of losses andwith full compensation. At the resonance, �0 = 1/LC , without compensation oflosses � = 0, but with full compensation � = ±�/2.

In Fig. 5.7 the currents and the voltages in the LC oscillator are represented bya phasor diagram, where represents the phase difference between the V1 and V2,and � is the phase difference between I1 and V1, which is

� = arctan

(gmc

gm

)

(5.21)

The circuit without mismatches is symmetric and this implies that there is perfectquadrature, = �/2, otherwise the voltages and currents would be different in thetwo oscillators, which is incompatible with the circuit symmetry.

5.4 Quadrature LC Oscillator with Mismatches

Without any mismatch there is total compensation of the losses, and the argument ofthe impedances R//L//C//g−1

m is ±�/2. In the case of two coupled oscillators withdifferent resonance frequencies (�01 and �02) we do not have full compensation oflosses, and there is a quadrature error. In the linear model of Fig. 5.5 the loop gain is:

G loop(s) = −g2mc

⎛

⎜⎜⎝

sL1

1 + sL1

(1

Rp1− gm1

)

+ s2 L1C1

⎞

⎟⎟⎠

⎛

⎜⎜⎝

sL2

1 + sL2

(1

Rp2− gm2

)

+ s2 L2C2

⎞

⎟⎟⎠

(5.22)

If we apply the Barkhausen criterion to the loop gain in equation (5.22), we donot arrive at a simple equation for the oscillation frequency. Thus, we will use adifferent approach to calculate the oscillation frequency.

In Fig. 5.8 we represent the oscillator phasor diagram when there are mismatches.With mismatches, there is a quadrature error �. The Barkhausen criterion im-

poses that 1+2 = � (1 = �/2+�, and 2 = �/2−�). The impedance phaseangles, �1 and �2, with mismatches (represented in Fig. 5.8) can be expressed as

�1 = arctan

(

gmc2V2 sin(1)

gm1V1 + gmc2V2 cos(1)

)

(5.23)

�2 = arctan

(

gmc1V1 sin(2)

gm2V2 + gmc1V1 cos(2)

)

(5.24)


V2gmc2

V1gm

I1

θ

φ1

θ

V2gm

V1gmc1−

I2

V1

V2

θ1Δ

Δφ

θ2Δ

V1−

φ2

θ1

θ2

Fig. 5.8 Phasor diagram with mismatches

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

�1 = arctan

(gmc sin(1)

gm + gmc cos(1)

)

(5.25)

�2 = arctan

(gmc sin(1)

gm − gmc cos(1)

)

(5.26)

Note that equation (5.25) and (5.26) result in equation (5.21) if = �/2. In orderto calculate the quadrature error we can differentiate equations (5.25) and (5.26), todetermine the � as a function of ��1 = �1 − � and ��2 = �2 − �:

5.4 Quadrature LC Oscillator with Mismatches 91

� =1 +

(gmc

gm

)2

+ 2

(gmc

gm

)

cos(1)

(gmc

gm

)2

+(

gmc

gm

)

cos(1)

��1 (5.27)

� =−1 −

(gmc

gm

)2

+ 2

(gmc

gm

)

cos(1)

(gmc

gm

)2

−(

gmc

gm

)

cos(1)

��2 (5.28)

Considering that we are close to the quadrature ≈ �/2 and cos() ≈ 0, forsmall mismatches � is proportional to ��.

� ≈(

1 +(

gm

gmc

)2)

��1 = −(

1 +(

gm

gmc

)2)

��2 (5.29)

Note that the above study has several approximations and is only valid for smallmismatches. Equation (5.29) implies that ��1 = −��2 = ��, and allows us tocalculate the quadrature error � due to mismatches as a function of the couplingintensity. Increasing the coupling reduces the quadrature error, which is in accor-dance with [63].

An RLC circuit with high parallel resistance has a high quality factor, and asmall deviation from the resonance gives a significant phase variation (as shown inFig. 5.9). This means that in coupled LC oscillators with high Q resonators, smallmismatches produce a high quadrature error. This explains why the first quadratureoscillators with low Q integrated inductors had a good quadrature relationship [11],whereas more recent realizations with higher Q and similar mismatches in the tank(for example 0.5%), have a quadrature error of 2◦ or higher [23, 63, 64, 98].

Having obtained the relationship between �� and �, we are able to calculatethe oscillation frequency with mismatches.

Let us consider, for simplicity, that there is a variation of the resonance frequencyof the first oscillator, �01 = �0 +��0, and that the resonance frequency of the other

Fig. 5.9 Phase of the tankimpedance for different Qs

ω

2

π

2

π−

ω0

High Q

LowQ

[ R //L // C ]θ = arg


oscillator is not affected, �02 = �0. The oscillation frequency changes, becoming�osc +��osc, and the phase of both oscillators will change: ��1 = �1 −� and ��2 =�2 − �. The phase of the first oscillator has a change due to mismatches and anotherdue to the variation of the oscillator frequency, i.e., ��1 = f (��0, ��osc), whilethe phase of the second oscillator is only affected by the change on the oscillationfrequency, i.e., ��2 = f (��osc); we have the relationship ��1 = −��2 = ��.

The impedance phase of an RLC circuit at �osc is

�(�osc) = �

2− tan−1

L�osc1

Rp(

1 −(

�osc

�0

)2) (5.30)

and

�1(�osc + ��osc) = �

2− tan−1

L1(�osc + ��osc)1

Rp1(

1 −(

�osc + ��osc

�0 + ��0

)2) (5.31)

�2(�osc + ��osc) = �

2− tan−1

L(�osc + ��osc)1

Rp2(

1 −(

�osc + ��osc

�0

)2) (5.32)

Using equations (5.30) and (5.31) we determine ��1 = �1 − �. With equa-tions (5.30) and (5.32) we determine ��2 = �2 − �. Knowing that ��1 = −��2, wecan obtain the oscillation frequency, but this leads to complicated equations (it maybe convenient to use a symbolic analysis program).

5.5 Q and Phase-Noise

Equation (5.5) defines the tank quality factor. In this section we will analyse theinfluence on Q of deviations of the resonance frequency.

The impedance phase of an RLC circuit is

�(�) = π

2− tan−1

L�1

Rp

(1 − LC�2)(5.33)

and

5.5 Q and Phase-Noise 93

d�(�)

d�= L RP (C L�2 + 1)

C2L2 R2p�4 − 2C L Rp

2�2 + L2�2 + Rp(5.34)

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

Q = �0

2

∣∣∣∣∣

L RP (C L�2 + 1)

C2L2 R 2p �4 − 2C L Rp

2�2 + L2�2 + Rp

∣∣∣∣∣

(5.35)

At the resonance frequency, (5.34) simplifies to:

d�(�)

d�= −2C RP (5.36)

and Q is given by (5.6), as expected.A maximum of 3 dB improvement of the oscillator phase-noise can exist in two

coupled LC oscillators when compared with a single LC oscillator [62–64, 96]. Inorder to take into account this improvement, we introduce a factor 1/2 in (5.9),

L{��} = 10 · log

[

1

2

2k FT

Pcarrier

(�0

2Q��

)2]

(5.37)

This maximum 3 dB improvement, due to coupling, is obtained when the oscilla-tors are isolated and oscillate at their common resonance frequency. When the oscil-lators are coupled, the oscillation frequency changes according to (5.17) or (5.18),and the theoretical equation (5.35) for Q should be used. From (5.35) and (5.37) weconclude that cross coupling two LC oscillators leads to phase-noise degradation,due to the reduction of Q (even without mismatches).

To illustrate the effect of a frequency shift on Q and phase-noise we consider,as an example, an RLC circuit with a resonance frequency of 5 GHz and differentvalues of Q0 (value of Q at the resonance frequency): Q0 = 5 (typical value ofintegrated inductors), Q0 = 10 (typical value of high performance integrated induc-tors), and Q0 = 30 (typical value for external inductors or for integrated inductorsin recent technologies with special RF options). Figs. 5.10 and 5.11 show the theo-retical plots of Q and of the phase-noise degradation as a function of frequency.

In Figs. 5.10 and 5.11 it is clear that, with high Q inductors, as the frequency ofoscillation deviates from the nominal value of 5 GHz, Q degrades steeply, and thephase-noise degrades accordingly. It can also be observed that the highest the Q, themore steeply the phase-noise degrades.

The most important conclusion from this study is that the performance of singleLC oscillators is different from the performance of coupled LC oscillators. Sin-gle LC oscillators oscillate at the resonance frequency and have good phase-noiseperformance, being the right choice for applications with very stringent phase-noisedemands (e.g. GSM [99,100]). Coupled LC oscillators can have a theoretical phase-noise improvement of 3 dB, but, any deviation from the resonance frequency due


−1000 −800 −600 −400 −200 0

Q0 = 30

Q0 = 10

Q0 = 5

200 400 600 800 10000

5

10

15

20

25

30

Offset from resonance frequency [MHz]

Qu

alit

y fa

cto

r

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

0 100 200 300 400 500 600 700 800 900 10000

5

10

15

20

25

30

35

40

45

Offset from resonance frequency [MHz]

Ph

ase-

no

ise

deg

rad

atio

n [

dB

]

Q0 = 30

Q0 = 10

Q0 = 5

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

5.5 Q and Phase-Noise 95

to mismatches and to coupling will reduce Q and increase the phase-noise; thiseffect is more severe for high Q resonators. Thus, for coupled LC oscillators to takeadvantage of Q enhancement provided by modern technologies, it is mandatory thatthe process mismatches are low.

In order to demonstrate the study above, a quadrature LC oscillator (Fig. 5.4) wasdesigned, for an oscillation frequency of 5 GHz, using the same 0.18 m CMOStechnology that was used in the example of a quadrature RC oscillator in Chap-ter 3. The coupled oscillators have the following circuit parameters: (W/L) =100 m/0.18 m for M transistors, (W/L) = 100 m/0.18 m for MSL transistors,Q0 = 10, I = 1 mA, and a supply voltage of 1.8 V.

Figure 5.12 shows the simulated phase-noise for a stand-alone LC oscillator andfor the cross-coupled LC oscillator. In a stand-alone LC oscillator the phase-noiseis −131.65 dBc/Hz @ 10 MHz. With weak coupling the quadrature oscillator has a3 dB improvement when compared with the single oscillator. With strong couplingthe phase-noise increases to −123.90 dBc/Hz @ 10 MHz. This is due to the fre-quency shifts originated by the coupling, which means that the oscillators oscillateat a frequency for which the Q value of each oscillator is lower than that for theresonance frequency �0.

In Chapter 7 we will present a comparison of the quadrature LC oscillator in thisexample with the quadrature RC relaxation oscillator of the example in Chapter 3.The oscillators in the two examples have the same oscillation frequency (5 GHz)and have been designed with the same technology.

104 105 106 107 108–160

–140

–120

–100

–80

–60

–40

Offset Frequency [Hz]

Pha

se N

oise

[dB

c/H

z]

1) Stand alone2) ISL = 0.4 mA 3) ISL = 3 mA

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


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

5.6 Quadrature LC Oscillator-Mixer

In this section we will not perform a theoretical study of the LC oscillator-mixer.We will only show that it is possible to perform mixing in the LC cross-coupledoscillator and we will compare the results with the RC oscillator.

To evaluate the possibility of performing the mixing function in the quadratureLC oscillator we insert an input signal in the feedback loop using the same ap-proach used for the cross-coupled relaxation oscillator: the soft-limiter tail current(iSL ) has a modulating signal, in addition to a DC component. The circuit usedto perform the simulations is the cross-coupled LC oscillator of Fig. 5.4 in AMS0.35 m CMOS technology, with a supply voltage of 2 V. The circuit parametersare (W/L) = 50 m/0.35 m for M transistors, (W/L) = 50 m/0.35 m for MSL

transistors, I = 2 mA, and ISL = 1 mA. For the resonators we use L = 10 nH andC = 420 fF (including parasitics) to achieve an oscillation frequency of 2.4 GHz,with a resonator Q = 5.

We have used iSL = 1 + 0.5 cos(�1t) [mA] with f1 = 300 MHz. In Fig. 5.13we observe the quadrature outputs in the time domain, modulated by the injected

20 21 22 23 24 25 26 27 28 29 30−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Time (ns)

Am

plit

ud

e (V

)

v1 v2

Fig. 5.13 Quadrature outputs with modulation

5.6 Quadrature LC Oscillator-Mixer 97

20 20.1 20.2 20.3 20.4 20.5 20.6 20.7 20.8 20.9 21−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Time (ns)

Am

plit

ud

e (V

)v1

v2

Fig. 5.14 Quadrature modulated outputs (zoom)

105 106 107 108–160

–150

–140

–130

–120

–110

–100

–90

–80

Offset frequency (Hz)

Pha

se n

oise

(dB

c/H

z)

Oscillator

Oscillator-Mixer

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


signal (with a gain of 50 mV/0.5 mA), where the modulation index is 0.05 (50 mVof modulation signal for 1 V of amplitude).

In Fig. 5.13 we can observe the modulated signals, but we can not evalu-ate whether the oscillator outputs remain accurately in quadrature. The zoom inFig. 5.14 shows that the oscillator outputs are in quadrature.

It should be noted that the mixing results from the circuit non-linearity, and anLC oscillator is, typically, a quasi-linear circuit; furthermore, the LC tank acts asa filter that strongly attenuates any signal with frequency different from the centralfrequency. Thus, this LC oscillator-mixer exhibits a low modulation index, whencompared with the relaxation oscillator-mixer.

The phase-noise plot for the oscillator with and without modulation is shown inFig. 5.15. We observe that the modulation leads to a slight degradation in the phase-noise for low offset frequencies. The simulated phase-noise is −135.9 dBc/Hz @10 MHz (this simulation is done with spectreRF considering ideal current sources).The expected result from equation (5.37) at the same offset is −134.5 dBc/Hz @10 MHz using F = 2, which is a typical value for this parameter. Recently somework was done concerning the exact determination of the factor F, considering thenoise from the transistors and the current source [43].

5.7 Conclusions

Single LC oscillators are widely used due to their good phase-noise performance.In this chapter we show that there is phase-noise degradation if two LC oscillatorsare coupled to obtain a quadrature oscillator. It is shown that to obtain good phase-noise performance, the technology used should provide high Q inductors and goodmatching. Otherwise, the good phase-noise performance of a single LC oscillator,will be lost by the effect of mismatches when two LC oscillators are coupled.

We present simulations of a quadrature LC oscillator, at 5 GHz, in which westudy the effect of coupling on the oscillator phase-noise. Single LC oscillators os-cillate at the resonance frequency and have low phase-noise. We observe that withweak coupling (with a negligible shift of the oscillation frequency with respect tothe resonant frequency of individual oscillators) there is a reduction in the oscillatorphase-noise of about 3 dB. However, the deviation from the resonance frequency dueto strong coupling (necessary for low quadrature error as shown in equation (5.29)),produces a significant degradation of the quadrature LC oscillator phase-noise.

We show that mixing can be done in LC quadrature oscillators and that theoutputs remain accurately in quadrature. However, an LC oscillator is tuned to asingle frequency, and the injection of any signal at a different frequency is stronglyattenuated by the LC tank, which leads to a low modulation index.

Chapter 6Two-Integrator Oscillator

Contents

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.2 High Level Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.2.1 Non-Linear Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006.2.2 Quasi-Linear Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.3 Circuit Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.4 Phase-Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.6 Two-Integrator Oscillator-Mixer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.6.1 High Level Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.6.2 Circuit Implementation and Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.1 Introduction

Integrated wireless systems capable of operating with different frequency bands anddifferent telecommunications standards are of great interest. In this chapter we pro-pose a quadrature oscillator that is able to operate in a wide range of frequencies,and which can also perform the mixing function.

In previous chapters we studied the quadrature relaxation oscillator, which isstrongly non-linear, and the quadrature LC oscillator, which is quasi-linear. Theoscillator studied in this chapter is different from the two previous ones because itdoes not have coupling: it is a single oscillator with inherent quadrature outputs.Both relaxation and LC oscillators have a limited tuning range, which is below onedecade (typically lower than 20%). The motivation for the study of this third type ofoscillator is to achieve a wideband quadrature oscillator.

The circuit described in this chapter has two-integrators in a feedback structure.It is an RC oscillator, but it is different from the circuit of Chapter 3: here, instead ofthe memory block (Schmitt-trigger) there is a second integrator. It is possible to havea higher oscillation frequency and wide tuning range (about a decade). We showthat this oscillator has an interesting characteristic: it can have either a non linearbehaviour (similar to relaxation oscillators) or a quasi-linear behaviour (typical ofLC oscillators).


99

100 6 Two-Integrator Oscillator

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

6.2 High Level Study

As we have done in previous chapters we consider a high level model of the two-integrator oscillator, in order to clarify the basic principles involved in its operation.

6.2.1 Non-Linear Performance

The oscillator in Fig. 6.1 is composed of two-integrators and two hard-limiters thatimplement the sign function, connected in a feedback loop. Each integrator outputdetermines the input polarity of the other integrator [8]. The oscillation frequencyis proportional to the integrators’ constant, and depends on the oscillator amplitude.The waveforms are rectangular at the hard-limiter outputs, and are triangular at theintegrators outputs. The two-integrator outputs are represented, with the normalizedvalue of 1, in Fig. 6.2.

To analyse the oscillator performance we consider that in the circuit of Fig. 6.1the integrator constants are equal and that for t = 0 the initial values of the inte-grator outputs are different: 1 for integrator 1 and 3 for integrator 2 (Fig. 6.3). Theoscillation amplitude VOUT is the sum of the two initial values.

The oscillation frequency and the amplitude are related by:

f0 = Ki

2VOUT(6.1)

where Ki is the integration constant, and VOUT is the output amplitude.With different integration constants, Fig. 6.4 shows that the amplitudes of the

integrator outputs are different, but the outputs have the same frequency and areaccurately in quadrature.

The integrator output amplitudes are dependent on the initial conditions of two-integrators and on their integration constants:

Fig. 6.1 Two-integratoroscillator with hard-limiters

vOUT1 vOUT2

1

–1

1 2 2


Time

vOUT1 vOUT2

1A

mp

litu

de

Fig. 6.2 Two-integrator oscillator triangular integrator outputs

Time

vOUT1 vOUT2

4

Am

plit

ud

e

Fig. 6.3 Integrator outputs with equal integrator constants


Time

vOUT1

vOUT2

Am

plit

ud

e

Fig. 6.4 Integrator outputs with different integrator constants

VOUT1 = 2

(

VINT1 + Ki1

Ki2VINT2

)

(6.2)

VOUT2 = 2

(

VINT2 + Ki2

Ki1VINT1

)

(6.3)

where VINT1 and VINT2 are the initial values.To determine the oscillation frequency we can substitute equations (6.2) or (6.3)

in (6.1). The oscillation frequency is:

f0 = 1

4(

VINT1Ki1

+ VINT2Ki2

) (6.4)

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

6.2.2 Quasi-Linear Performance

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


vOUT1 vOUT2

1 1

–1

2 2

Fig. 6.5 Two-integrator oscillator with soft-limiters

oscillator amplitude does not saturate the soft-limiter, the oscillator has a linear be-haviour, with sinusoidal outputs (Fig. 6.6), and the oscillator quadrature relationshipis preserved.

With this approach the circuit is linear, and the loop gain is:

(K1 K2

s2

)

= G loop(s) (6.5)

Using the Barkhausen criterion, the loop gain (6.5) is equal to 1. Solving (6.5) inorder to � we obtain the oscillation frequency:

�0 =√

K1 K2 (6.6)

Time

vOUT1vOUT2

Am

plit

ud

e

Fig. 6.6 Two-integrator oscillator with sinusoidal outputs


Time

vOUT1

vOUT2

Am

plit

ud

e

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

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

6.3 Circuit Implementation

A two-integrator oscillator circuit is presented in Fig. 6.8. Each integrator is realizedby a differential pair (transistors M) and a capacitor (C). The oscillator frequency iscontrolled by Itune. There is an additional differential pair (transistors ML ), with theoutput cross-coupled to the inputs, which performs two related functions:

� compensation of the losses due to R to make the oscillation possible (a negativeresistance is created in parallel with C);

� amplitude stabilization, due to the non-linearity (the current source Ilevel controlsthe amplitude);

It should be noted that the correspondence between the circuit of Fig. 6.8 and theblock diagrams in the previous section (Figs. 6.1 and 6.5) is conceptual and nottopological: the integrators with limited output shown in Fig. 6.8 are modeled bythe ideal integrators in cascade with limiters.

The circuit of Fig. 6.8 can be represented by the linear model in Fig. 6.9, wherethe negative resistance is realized by the cross-coupled differential pair (ML ), and Rrepresents the integrator losses due to the pairs of resistances R/2.

6.3 Circuit Implementation 105

Itune

Ilevel

Itune

Ilevel

ML ML

M M M M

ML ML

VCC

C1 C2

vOUT1 vOUT2

R/2 R/2 R/2R/2

Fig. 6.8 Two-integrator oscillator implementation

From the model in Fig. 6.9, which is valid for quasi-linear performance, we canobtain the oscillator frequency using the loop gain of the oscillator. For oscillation,the losses must be compensated (Rp = 1/gmL ), each stage is a perfect integrator,and the phase condition is achieved for all frequencies, because each stage of thetwo-integrator oscillator gives a 90◦ phase shift, as required for quadrature outputs.Thus, the two-integrator oscillation frequency is determined by the amplitude con-dition. The loop gain is:

|H ( j�)| = g2m

�2C2(6.7)

Fig. 6.9 Two-integratoroscillator linear model

1gmL

C R

C R

gm

–gm

–

1gmL

–


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

�0 = gm

C(6.8)

From equation (6.8) we can conclude that the oscillator frequency varies bychanging either the capacitance or the transconductance. In a practical circuit wecan use varactors to change the capacitance or, most commonly, we can changethe tuning current and therefore the transconductance. With the second approach, ifthe transconductances are implemented by bipolar transistors the frequency changeslinearly with the tail current. If the transconductances are implemented with MOStransistors the frequency will be proportional to the square root of the tail current.Since we can change the transconductance in a wide range, these oscillations havewide tuning range.

The circuit of Fig. 6.8 can work in two different modes:(1) If we over-compensate the losses by increasing Ilevel , the performance is non-

linear and resembles that of the block diagram in Fig. 6.1. With a strong non-linearperformance (the transistors operate as switches) the waveforms are approximatelytriangular. In this case the oscillator amplitude is:

VOUT∼= Ilevel R (6.9)

and using (6.1) we obtain the oscillator frequency:

f0 = Itune

2CVOUT(6.10)

In this case the oscillator has a behaviour similar to that of a relaxation oscillator.(2) If we compensate the losses only to the amount necessary for the oscillations

to start, the circuit of Fig. 6.8 is modeled by that of Fig. 6.9. The transistors workin the linear region, and the outputs are close to sinusoidal with the amplitude thatsatisfies the condition,

1

gmL= R (6.11)

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

VOUT∼= Ilevel R (6.12)

6.4 Phase-Noise 107

Fig. 6.10 Differential voltageto current transfercharacteristic of adifferential pair

−Itail

Itail

iD1 – iD2

v1 – v2

6.4 Phase-Noise

In this section we consider that the oscillator has a quasi-linear behaviour (sinusoidaloutputs), and we calculate the oscillator phase-noise using the approach for linearoscillators [36]. In the following derivation we consider the oscillator as a feedbacksystem, and we represent the effect of all noise sources by one equivalent noisesource at the input, as shown in Fig. 6.11. The objective of the following analysis isto calculate the noise transfer function of the oscillator (Leeson’s equation).

The closed-loop transfer function is

N (s) = Yn(s)

Xn(s)= H (s)

1 − H (s)(6.13)

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

H ( j�) ≈ H ( j�0) + �md H

d�

∣∣∣∣�=�0

(6.14)

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

N (�m)

Fig. 6.11 Linear oscillatorwith noise input

xn ynH(s)

Noise source


N (�m) = 1 + �md Hd�

−�md Hd�

(6.15)

In practical cases |�md H/d�| << 1 [1, 36], so

N (�m) ≈ −1

�md Hd�

(6.16)

and the noise power spectral density at � = �0 + �m is

S(yn) = |N (�m)|2S(xn) (6.17)

with

|N (�m)|2 = 1

(�m)2(

d Hd�

)2 (6.18)

Equation (6.18) shows that at � = �0 + �m the output noise spectral density, ismultiplied by −(�md H/d�)−2, as shown in Fig. 6.12.

We will now use the oscillator quality factor in the noise equation. The oscillatorquality factor is defined as [36]:

Q = �0

2

√(

d A

d�

)2

+(

d�

d�

)2

(6.19)

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

d H

d�=(

d A

d�+ j A

d�

d�

)

exp( j�) (6.20)

Fig. 6.12 Oscillatorphase-noise [36] ω ωω0 ωω0

N (ω)2

Input noisespectral density

Output noisespectral density

6.4 Phase-Noise 109

Equation (6.18) can be rewritten as,

|N (�m)|2 ≈ 1

(�m)2[(

d Ad�

)2 + A2(

d�d�

)2] (6.21)

Noting that A ≈ 1 in steady-state oscillations, from (6.19) and (6.21), we obtain

|N (�m)|2 ≈ 1

4Q2

(�0

�m

)

(6.22)

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

H ( j�) = �20

�2(6.23)

and

d�

d�= 0 (6.24)

d A

d�= −2

�20

�3(6.25)

At the oscillation frequency equation (6.25) simplifies to:

d A

d�

∣∣∣∣�=�0

= − 2

�0(6.26)

and from (6.19) the two-integrator oscillator quality factor at the oscillation fre-quency is:

Q0 = �0

2

√(

2

�0

)2

+ (0)2 = 1 (6.27)

The value of Q is unity, which is lower than the typical values for an LC oscillator.This explains the poor performance in terms of phase-noise of the two-integratoroscillator when compared with the LC oscillator.

We consider now that all noise sources are represented by two independent cur-rent noise sources in1 and in2 as shown in Fig. 6.13.


Fig. 6.13 Oscillator withnoise current sources

CR

C R

gm

in1

in2

1

gmL1−

1gmL2

−

−gm

ino

Using (6.23) and (6.13) the noise response for a noise current in1 is

N1(�m) =�2

0�2

1 − �20

�2

(6.28)

Using the approximation of (6.14)

�20

�2≈ 1 − �m

2

�0(6.29)

and (6.28) becomes

N1(�m) ≈ 1

1 − 1 − �m2

�0

= − �0

2�m(6.30)

It can easily be shown that in the circuit of Fig. 6.13 the two noise sources in1

and in2 have the same effect on ino if we are close to the resonance frequency. Usingthis approximation, the total noise power density, due to in1 and in2, is

S[ino(�m)] = 1

4

(�0

�m

)2

[S(in1) + S(in2)] (6.31)

where S(in1) = S(in2) = 4kT/R, if we consider only the thermal noise of resis-tors R.

Equation (6.31) includes both amplitude and phase-noise, and since L(�m) onlytakes into account the phase-noise an extra factor 1

2 [67] should be included. Usingequation (6.31) the two-integrator oscillator phase-noise is

L(�m) = 10 log

(12 S[inout (�m)]

I 2rms

)

= 10 log

[

4kT

RI 2rms

(�0

2�m

)2]

(6.32)

6.5 Simulation Results 111

where Irms is the rms current at the output of transconductance gm in Fig. 6.13.It should be noted that (6.32) is only valid for the 1/ f 2 region of Fig. 2.15; for

the 1/ f 3 region we need to add to (6.32) the contribution of the flicker noise of theactive devices.

6.5 Simulation Results

The circuit of Fig. 6.8 was designed for the 0.35 m technology of AMS. InFig. 6.14 a plot of the circuit layout is shown. All the simulations are done withspectreRF, using RF models for the circuit components. The total circuit area is500 × 400 m2, of which 150 × 100 m2 corresponds to the oscillator (the cir-cuit area is dominated by the pads). The circuit parameters are: R = 400 �,(W/L) = 160 m/0.35 m for M transistors, (W/L) = 80 m/0.35 m for ML

transistors, C ≈ 200 fF, and Ilevel = 2 mA. The supply voltage is 3 V.We obtain an extended tuning range, 900 MHz to 5.8 GHz, by changing the tun-

ing current, Itune. The circuit outputs are close to sinusoidal (one of the two casesin the high level study). The frequency is proportional to the transconductance ofthe MOS differential pair, which, in turn, is proportional to the square root of thecurrent. This explains the shape of the curve of frequency versus coupling currentin Fig. 6.15.

Figure 6.16 shows that if we change the integrator constants by 10%, the oscilla-tor amplitudes change, but the oscillator outputs remain in quadrature (the quadra-ture error is 2.1◦). This is in accordance with the theoretical analysis (Fig. 6.7).

We obtain by simulation the phase-noise for different oscillation frequencies(Table 6.1), and we compare the results with those predicted by equation (6.32)in which Irms is obtained by simulation. In Table 6.1, these results are obtained with

Fig. 6.14 Circuit layout


Fig. 6.15 Frequency vstuning current

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7 8 9

Itune (mA)

Freq

uenc

y (G

Hz)

10

16 17 18 19 20 21 22 23−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Time (ns)

vOUT1

vOUT2

Am

plit

ud

e (V

)

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

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

Tuning current Irms Frequency Eq. (6.32) Simulated

836 A 580 A 900 MHz −120.14 dBc/Hz −118.98 dBc/Hz1.658 mA 1.20 mA 1.8 GHz −120.31 dBc/Hz −121.62 dBc/Hz1.87 mA 1.30 mA 2.0 GHz −120.09 dBc/Hz −121.53 dBc/Hz2.19 mA 1.55 mA 2.4 GHz −120.03 dBc/Hz −121.07 dBc/Hz8.27 mA 3.70 mA 5.8 GHz −119.93 dBc/Hz −117.50 dBc/Hz

6.6 Two-Integrator Oscillator-Mixer 113

R = 400 �, k = 1.38 × 10−23 (J/K) and T = 300 K, and with the Irms values (atthe transconductance output) indicated in the table.

In Table 6.1 we observe that the oscillator phase-noise is approximately constantover the tuning range and has worse values at the lower and upper ends. We observethat there is a good agreement between the simulation results and those obtainedusing equation 6.32.

Although this is a quasi-linear oscillator, the phase-noise is similar to that of arelaxation oscillator and higher than the phase-noise of an LC oscillator. This is dueto the higher quality factor of the LC oscillator, which is typically higher than 1,while the two-integrator oscillator has a quality factor of 1, as shown above.

6.6 Two-Integrator Oscillator-Mixer

6.6.1 High Level Study

The modulating signals can be injected either before or after the integrator blocks.Injecting the signal before an integrator will originate FM modulation; therefore weinject the modulation signal after the integrator. In Fig. 6.17 the block diagram ofthe two-integrator oscillator-mixer is presented.

The oscillator signals remain in quadrature after the injection of a modulatingsignal and the mixing function is performed. The modulating signal must be alwayspositive, which means that it must have a DC component. In the case of sinusoidalmodulating signal, we would have

v1 = v2 = A + a cos(�1t) (6.33)

where |a| < A.In Fig. 6.18 a high level simulation of the oscillator-mixer is presented, in which

the modulating signal has amplitude 0.3 V, with a DC component of 1 V, and fre-quency f1 = 0.1 f0.

vOUT1v1 vOUT2v2

1 1 2 2

–1

Fig. 6.17 Two-integrator oscillator-mixer block diagram


0 5 10 15 20 25 30 35 40 45 50−1.5

–1

−0.5

0

0.5

1

1.5

Time (ns)

Am

plit

ud

e (V

)

Fig. 6.18 Two-integrator oscillator-mixer waveforms

6.6.2 Circuit Implementation and Simulations

An oscillator-mixer was designed with the circuit of Fig. 6.8, and we apply themodulating signal to the current sources Ilevel , since we want to modulate the outputamplitude (modulation of Itune would change the oscillation frequency, which mightbe a good solution to produce frequency modulation).

It should be noted that there is not a complete correspondence between the circuitand the high level block diagram. At circuit level the amplitude stabilization basedon a negative gmL (which is not present in the high level model) is used for themodulation. In spite of this difference, we believe that the study in terms of thehigh-level model is useful to highlight some of the basic concepts involved.

The results in Figs. 6.19, 6.20, and 6.21 are obtained with ilevel = 2+0.5 cos(�1t)[mA] with f1 = 300 MHz. In Figs. 6.20 and 6.21 we represent the oscillator outputsvOUT1 and vOUT2, in the time and in the frequency domains, which confirm that thistwo-integrator oscillator can be used as a combined oscillator-mixer.

The output voltage is

vOUT∼= RIlevel (1 + x(t)) cos(�0t) (6.34)

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

6.6 Two-Integrator Oscillator-Mixer 115

10 11 12 13 14 15 16 17 18 19 20−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8vOUT2vOUT1

Time (ns)

Am

plit

ud

e (V

)

Fig. 6.19 Oscillator-mixer output in the time domain

10 10.1 10.2

0.8

10.3 10.4 10.5 10.6 10.7 10.8 10.9 11−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

vOUT1vOUT2

0.6

Time (ns)

Am

plit

ud

e (V

)

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


0 1 2 3 4 5 6−70

−60

−50

−40

−30

−20

−10

0

Frequency (GHz)

f0 +

f1f0

– f1

Am

plit

ud

e (d

BV

)f0

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

105 106 107 108–150

–140

–130

–120

–110

–100

–90

–80

–70


Oscillator-Mixer

Oscillator

Pha

se N

oise

(dB

c/H

z)

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

6.7 Conclusions 117

= |xmax|Ilevel

(6.35)

The simulated modulation index (Fig. 6.19) is about 0.22 (200 mV/900 mV),which is in accordance with (6.35). Comparing this result with those obtained forthe previous relaxation and LC oscillators, we can conclude that the present circuithas the highest modulation index.

The other mixer characteristics, noise factor, intermodulation distortion (IIP2 andIIP3), 1 dB compression point, and isolation between the two ports (LO leakage), arenot discussed here.

The oscillator phase-noise with and without modulating signal, represented inFig. 6.22, shows that the influence of the mixing in the oscillator phase-noise canbe neglected, as it happened in the RC and LC oscillators considered in previouschapters.

6.7 Conclusions

In this chapter we study a third type of quadrature oscillator: the two-integratoroscillator. The main advantage of this oscillator, when compared with relaxationand LC oscillators, is its wide tuning range, which in a practical implementation(GHz range) can be about one decade.

We presented a high-level study of a two-integrator oscillator in which we con-sider hard-limiting (triangular outputs) and soft-limiting (sinusoidal outputs). Inboth cases, mismatches cause a change of amplitude and oscillation frequency, butthe outputs remain in quadrature.

A circuit implementation was presented and an equation has been derived forthe oscillator phase-noise. The circuit has a wide tuning range, from 900 MHz to5.8 GHz, and is suitable for use in different applications. The circuit can work witha varying degree of non-linearity: if we compensate the losses significantly morethan it is necessary for oscillation, there is a strong non-linearity, and the behaviouris close to that of a relaxation oscillator, with triangular outputs. In our design thecompensation of the losses is near the minimum required for oscillation, and theoscillator behaviour is approximately linear, close to a second order oscillator withsinusoidal outputs.

This two-integrator oscillator can perform the mixing function throughout itswide tuning range, while preserving the quadrature relationship. This oscillator-mixer has the highest modulation index when compared with circuits considered inprevious chapters.

Chapter 7Measurement Results

Contents

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1197.2 Quadrature Relaxation Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7.2.1 Circuit Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1207.2.2 Measurement Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

7.3 Quadrature LC Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237.3.1 Circuit Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237.3.2 Measurement Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

7.4 Quadrature Oscillator-Mixer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277.4.1 Circuit Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277.4.2 Measurement Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

7.5 Comparison of Quadrature LC and RC Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1327.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

7.1 Introduction

In this chapter we present measurement results of five prototype circuits, in the GHzrange of frequencies, to validate the theoretical analysis concerning the quadratureerror and phase-noise, and to validate the oscillator-mixer concept.

We designed a quadrature relaxation oscillator at 2.4 GHz, and we confirm bymeasurements that strong coupling leads to a reduction of the phase-noise and de-creases the quadrature error.

We also designed a quadrature LC Oscillator at 1 GHz, and we confirm that,although single LC oscillators have low phase-noise, with strong coupling there isan increase of the phase-noise.

The objective of the third circuit prototype is to validate the oscillator-mixer con-cept at 5 GHz.

Two prototypes at 5 GHz are used in a comparative study of cross-coupled RCand LC oscillators.

In this chapter we describe only the circuit schematics, and we present the mea-surement results. The complete measurement setup and the procedure used to mea-sure the oscillator phase-noise and the quadrature error are described in appendix A.


119

120 7 Measurement Results

7.2 Quadrature Relaxation Oscillator

In this section we present the quadrature relaxation oscillator with an oscillationfrequency of 2.4 GHz. We present measurement results to show the influence of thecoupling gain on the quadrature relationship and phase-noise.

7.2.1 Circuit Schematic

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

M

R R R R

CM M M

MSL MSL MSL MSL

C

VCC

ISL

M2 M2 M2

I

M1 M1 M1 M1 M1

Fig. 7.1 Quadrature relaxation oscillator schematic


Fig. 7.2 Die photo of thequadrature RC oscillator

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

7.2.2 Measurement Results

Since a high precision quadrature relationship is expected, visual inspection of thewaveforms does not suffice. The phase difference between two sinusoidal signals xand y can be determined by using the following procedure.

If

x(t) = Ax sin(�0t + �) (7.1)

y(t) = Ay cos(�0t) (7.2)

where � represents the error in the quadrature relationship, then

2

Ax AynT

t0+nT∫

t0

x(t)y(t)dt = sin(�) ≈ � (7.3)


where T = 2�/�0 is the period and n is the number of periods. This method ofdetermining the quadrature error is robust with respect to additive noise.

As we explain in Chapter 2, the image rejection ratio (IRR) in the absence ofgain mismatch is [1]

I R R = �2

4(7.4)

The two quadrature outputs at 2.4 GHz, which have about 120 mVpp, are repre-sented in Fig. 7.3. To have the clean plot of Fig. 7.3 we have stored 16 samples ofthe signal, which are averaged to remove the oscillator noise. We stored the outputwaveforms truncated in such a way that an integer number of periods remain. Afterthat, any offset was removed, and the amplitude was normalized to 1. Finally, theabove procedure was used to obtain the quadrature error.

The measured quadrature errors were lower than 1◦ with strong coupling. Byinterchanging the output paths, length errors can be distinguished from circuit er-rors. If we reduce the coupling gain by changing the current to a weak value, themeasured error is higher than 4◦.

The oscillator frequency changes between 2.47 GHz, with weak coupling, to2.29 GHz, with strong coupling. Thus, by increasing the coupling gain, the accuracyin the quadrature relationship is improved, but the oscillator frequency is reduced.The results are summarized in Table 7.1.

To measure the oscillator phase-noise we use a spectrum analyzer. Since thephase-noise has units of dB with the respect to the carrier (dBc/Hz) we calculatethe noise power in a 1 Hz bandwidth and divide the result by the carrier power (in

Fig. 7.3 Oscillator quadrature outputs

7.3 Quadrature LC Oscillator 123

Table 7.1 Effect of coupling on the quadrature error for the oscillator in Fig. 7.1

ISL [mA] Frequency [GHz] Quadrature Error [◦]

1.17 (weak coupling) 2.47 4.31.92 2.4 2.23.12 (strong coupling) 2.29 0.8

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

Offset frequency Weak coupling Strong coupling

@600 kHz −93 dBc/Hz −100 dBc/Hz@1 MHz −97 dBc/Hz −105 dBc/Hz

this case −8 dBm). Since the spectrum analyzer bandwidth used in the measurementof the noise power is 20 kHz and

10 log10(20000) = 43 dB (7.5)

to have the phase-noise in dBc/Hz we need to subtract 35 dB (8 dB for the carrierand −43 dB due to the spectrum analyzer bandwidth) from the values measured inthe spectrum analyzer. In Table 7.2 the phase-noise is compared at two differentfrequency offsets from the carrier. The improvement of phase-noise with strongcoupling is about 7 dB for both frequency offsets.

It is very difficult to measure accurately the absolute value of the oscillatorphase-noise. The direct measurement with the spectrum analyzer has an accuracyof ±4 dB. To increase the accuracy we should use a phase-noise measuring sys-tem. However, the relative values are not strongly affected by the inaccuracy of themeasurement setup.

We can conclude that to have a cross-coupled relaxation oscillator with lowphase-noise and low phase error we should have a strong coupling gain, whichconfirms the study presented in Chapter 3.

7.3 Quadrature LC Oscillator

In this section we present the 1 GHz quadrature LC oscillator and the measurementresults for quadrature error and phase-noise.


The quadrature oscillator has been implemented using a cross-coupled LC oscillatortopology [11], shown in Fig. 7.4.

The implementation uses M and MSL transistors with (W/L) = 100 m/0.35 m,and M1 = M2 transistors with (W/L) = 30 m/1 m. The total capacitance value,including the parasitics from transistors and bonding pads is C ≈ 2 pF, and the


L

M M

L

C C

VCC

L

M M

L

C C

MSLMSL MSLMSL

IT

ISL

M1 M1

M2 M2

Fig. 7.4 Quadrature LC oscillator schematic

total bias current is 4 mA (2 mA for each oscillator). The oscillator outputs are con-nected to pads, and the inductance of the external inductor plus the bonding wireinductance is 9 nH. The circuit uses a 3 V supply. The overall area without pads is300 × 450 m2.

In this design we use an external inductor. Integrated inductors have some impor-tant drawbacks: they occupy a large area, and have either very low quality factor, orneed special RF options (in order to have reasonable quality factor). Since the objec-tive is to investigate the variation of phase-noise due to coupling, we use an externalinductor because it is simpler and has a much higher quality factor, necessary to putin evidence significant phase-noise degradation.

The external inductor used is a SMD inductor (manufactured by Coilcraft). Wechoose the 0805 size for facility of soldering this type of components. The 0805HQ series has the highest quality factors, and these ceramic chip inductors have 5%inductance tolerance and batch consistency. We use a 0805 HQ inductor with 6.2 nH@ 250 MHz, with a quality factor of 88 at 1 GHz.

The model of the inductor is shown in Fig. 7.5.The values of R1, R2, C1, and L are listed for each component type. We use a

0805HQ-6N2 circuit with R1 = 8 �, R2 = 0.04 �, C = 0.056 pF, and L = 6.4 nH.

7.3 Quadrature LC Oscillator 125

Fig. 7.5 External inductormodel

L

C1 R1

R2Rvar

The value of the frequency dependent variable resistor Rvar , depends on the skineffect and is calculated by:

Rvar = k√

f (7.6)

wheref is the frequency in Hz;k is a constant with the value of 1.15 × 10−05 for this particular case.This inductor has a self resonance frequency of 4.75 GHz, but, typically the self

resonance frequency of the component model will be higher than the value measuredon a circuit board, since the parasitic elements of the circuit board will lower theresonance frequency, especially for very small inductance values.

The total value of the inductance includes the contribution of the wirebonding.In Table 7.3 the characteristics of a typical bonding wire with a length of 2 mm areshown [105]. We use a wire with 1.0 mil1 diameter (values in bold in Table 7.3). Thelength is about 3 mm, which leads to an inductance of around 3 nH. The bond pitchin the circuit is higher than in Table 7.3, leading to a much lower mutual inductance.Adding the contributions of the external inductor and the wirebonding leads to atotal inductance of about 9 nH.

Again the circuit is implemented following general good layout practice[101–103], using detailed information on the AMS CMOS 0.35 m process [104].

The final circuit, represented in Fig. 7.6, has a total chip area of 550 × 700 m2. Asin any quadrature oscillator, the circuit layout must be as symmetrical as possible,and signal paths need to have the same length to avoid quadrature errors.

Table 7.3 Bonding parameters of bond wire with 2 mm (from [105]), for 90 m bond pad pitchand 180 m bond finger pitch

Bonding wire diameter 0.8 mil 1.0 mil 1.2 mil

Resistance (�) 0.154 0.103 0.079Inductance (nH) 2.089 1.996 1.915Capacitance (pF) 0.104 0.122 0.140Mutual inductance (nH) 0.9798 0.9787 0.9770

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


Fig. 7.6 Die photo of thequadrature LC oscillator


The oscillator outputs at 1.1 GHz, represented in Fig. 7.7, have amplitude of about250 mVpp. To have the clean plot of Fig. 7.7, we average 16 sampled waveformsto suppress the oscillator noise. We store the output waveforms, truncated to aninteger number of periods. After removing any offset the amplitude is normalizedto 1. Finally, the quadrature relationship is obtained by using equation (7.3).

The measured quadrature errors are about 5.6◦ over the tuning range (by inter-changing the outputs we remove the path length errors). If we increase the couplinggain (by increasing the bias current) the measured error is about 2.7◦. Therefore, ahigh coupling gain is necessary for the oscillator to have a lower quadrature error.

To measure the oscillator phase-noise we use the spectrum analyzer. The carrierpower is −2 dBm, and the spectrum analyzer bandwidth is 20 kHz. To obtain thephase-noise in dBc/Hz we need to subtract 41 dB (2–43 dB) to the values measuredby the spectrum analyzer.

In Table 7.4 the oscillator phase-noise for two different offsets is given. Withstrong coupling there is a degradation of more than 10 dB, and the oscillator fre-quency is reduced from 1.157 GHz to 1.051 GHz. This result is very important,because it confirms our theoretical analysis, which shows that the low phase-noiseof single LC oscillators is not maintained for strongly coupled LC oscillators.

7.4 Quadrature Oscillator-Mixer 127

Fig. 7.7 LC oscillator quadrature outputs

Table 7.4 Effect of coupling on the phase-noise

Offset frequency Weak coupling Strong coupling

@600 kHz −124 dBc/Hz −110 dBc/Hz@1 MHz −130 dBc/Hz −117 dBc/Hz

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

7.4 Quadrature Oscillator-Mixer

In this section we present the measurement results of a 5 GHz quadrature relax-ation oscillator-mixer. The main objective is to validate the oscillator-mixer concept.We also compare the phase-noise performance with that of other state-of-the artoscillators.


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


Fig. 7.8 Oscillator-mixerschematic

CR

biasI

CCV

R

Q

R

Q

1Q1Q

Q

1inLNA

SLQSLQ

R R

R

Q

R

Q

1Q1Q

2inLNA

SLQSLQ

R R

1C 1C

the oscillator-mixer is within the scope of this book, so the other blocks, are onlydiscussed in appendix A.

The oscillator-mixer has been implemented using the cross-coupled relaxationoscillator topology [8] shown in Fig. 7.8. The tail currents of the limiters in the crosscoupling paths are supplied by an LNA, so the gain of these limiters is modulatedby the LNA signal.

The implementation uses Q, Q1, and QSL transistors with emitter areas of0.32 m × 6 m, 0.32 m × 10 m, and 0.32 m × 6 m, respectively, R = 50 �,C1 = 300 fF, C = 2 pF, and the bias currents are 2 mA for the oscillator core

Fig. 7.9 Die photo of theoscillator-mixer

Low Pass Filters

Bias

LNA

OscMixDiv2


transistors and 2 mA for the tail currents of the soft-limiters. The input marked Ibias

in Fig. 7.8 is connected to a pad, and is used to tune the oscillator. The tuning rangeis around 20% fosc (1 GHz). All the transistors were dimensioned and biased tohave the highest fT possible in the process. The circuit uses a 2.5 V supply. Theimplemented circuit has an overall area of 250 × 100 m2.

The complete circuit was implemented following general layout good practicerules [101–103] and detailed information on the IBM BiCMOS6HP SiGe process[106]. The final circuit, represented in Fig. 7.9, has a total chip area of 2.0×2.4 mm2

(250 × 100 m2 for the oscillator-mixer). Again, we design the circuit layout assymmetrical as possible, and signal paths have the same length to avoid quadratureerrors.


The first measurement result concerns the oscillator-mixer tuning range. From thesimulations it was expected that the oscillation frequency would be between 5 and6 GHz. The measured frequency, between 4.28 GHz and 5.28 GHz, is lower by 10%.This difference can be explained by last minute changes in the circuit layout tofulfil requirements of the particular chip fabrication run used (mainly changes in theavailable capacitors). The measured tuning range agrees with the simulated value ofabout 1 GHz.

For the validation of the oscillator-mixer concept we include this circuit in a PLL,as described in appendix A. The oscillator is adjusted for 5 GHz, and the availableoutput is the output of the frequency divider by two, inside the PLL, and not the5 GHz oscillator output. In Fig. 7.10 the spectrum of the divider output at 2.5 GHzis shown, which is in accordance with simulations.

The LNA has a single-ended input and two equal current outputs, which con-trol the two soft-limiters in the quadrature RC oscillator. We inject a signal withamplitude −30 dBm and 5.010 GHz at the LNA input; this frequency has 10 MHzdifference form the oscillation frequency, which is the lowest possible frequency fora Low-IF in these 5–6 GHz bands, in which the channels have 10 MHz bandwidth.We expected an output signal of about −30 dBm, leading to an overall gain of 0 dB.However, the flip-chip gold pads used in the simulation, were later found not tobe available in the particular chip fabrication run used, leading to last minute useof wirebond pads in the circuit layout. The LNA became unmatched, leading to aLow-IF output amplitude of −50 dBm (1 mV in a 50 � load), as shown in Fig. 7.11(instead of the expected −30 dBm).

The quadrature errors, obtained by using the procedure described above (in 7.2.2)are of the order of 1◦ (Fig. 7.11) in 4 measured die samples.

A plot of the measured phase-noise is shown in Fig. 7.12. Since this noise hasbeen measured after frequency division by two, 6 dB must be added to find theactual phase-noise of the oscillator-mixer (assuming a noiseless frequency divider).Phase-noise close to the carrier is suppressed by the PLL, so it is the PLL reference


2.45 2.46 2.47 2.48 2.49 2.5 2.51 2.52 2.53 2.54 2.55–80

–70

–60

–50

–40

–30

–20

–10

0

Frequency [GHz]

Am

plitu

de [

dBm

]

Fig. 7.10 Divider by two output at 2.5 GHz

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


PLLreferenceoscillator

quadratureoscillator

Fig. 7.12 Oscillator phase-noise

oscillator that defines the output phase-noise within the PLL bandwidth, which isabout 100 kHz. Above 1 MHz, a slope of −20 dB/decade is observed, which is dueto the quadrature oscillator, and this is the phase-noise that we wish to measure.The phase-noise of the oscillator-mixer circuit (functioning as an oscillator only) at10 MHz offset from the carrier (at 5 GHz) is about −114 dBc/Hz.

The phase-noise was measured with an HP phase-noise measurement system,which has a better accuracy than a spectrum analyzer (the equipment used has anaccuracy of ±2 dB). Using this equipment we do not need to calculate the carrierpower and to take into account the spectrum analyzer bandwidth.

For a comparative study between this oscillator and other oscillators presented inthe literature, we use the following FOM (figure-of-merit) [107]:

FOM = Lmeasured + 10 log

((� f

f

)2 PDC

Pref

)

(7.7)

wherePDC is the DC power dissipated by the oscillator;Pref is a reference power level (typically 1 mW).This FOM contains information on phase-noise, oscillator frequency, offset fre-

quency, and power dissipation. The results in Table 7.5 show that the proposedquadrature oscillator performance is comparable to that of other state-of-the-art RCoscillators.

This test-circuit used without modulation is used to make a comparison withother state-of-the-art RC oscillators (either single or cross-coupled). The reason touse this circuit (and not those in Sections 7.2 or 7.5) is that we had access to a moreprecise measurement system for this circuit (at TU Delft).


Table 7.5 Comparison of state-of-the-art RC oscillators

Ref. Lmin [m] Vcc [V] f [GHz] � f[MHz]

L(� f )[dBc/Hz]

FOM[dBc/Hz]

fmax

[GHz]

fmax

fmin

[108] 1.2 5.0 0.93 0.1 −83 −154 0.93 2.9[109] 0.6 3.0 0.90 0.6 −117 −165 1.20 1.6[36] 0.5 3.0 2.20 5.0 −109 −158 2.20 –[36] 0.5 3.0 0.92 5.0 −102 −151 0.92 –[110] 0.35 3.3 0.97 1.0 −117 −158 0.97 –[37] 0.25 2.5 1.33 1.0 −112 −164 1.33 –[37] 0.25 2.5 5.43 1.0 −99 −154 5.43 1.3[111] 0.18 1.8 3.52 4.0 −106 −153 3.52 35This work 0.25 2.5 5 10 −114 −154 5.3 1.23

7.5 Comparison of Quadrature LC and RC Oscillators

In order to verify the conclusions concerning the phase-noise of RC oscillators and,in addition, to compare its noise with that of LC oscillators, we designed an RC andan LC oscillator for the same frequency (5 GHz), using the same technology (TSMCCMOS 0.18 m), and we had them fabricated on the same chip.

The quadrature RC oscillator, with the schematic of Fig. 7.1, has R = 100 �,(W/L) = 75 m/0.18 m for the M transistors, (W/L) = 100 m/0.18 m for theMSL transistors, C = 300 fF, I = 3 mA, and ISL = 500 A (weak coupling) orISL = 3 mA (strong coupling). The supply voltage is 3 V.

The quadrature LC oscillator has the circuit schematic of Fig. 7.4, and uses inte-grated inductors. It has (W/L) = 75 m/0.18 m for the M transistors, (W/L) =75 m/0.18 m for the MSL transistors, L = 2 nH, Q0 = 5, I = 1 mA, and

RC Oscillator LC Oscillator

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

7.5 Comparison of Quadrature LC and RC Oscillators 133

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

Coupling RC LC

Sim. Meas. Sim. Meas.

Weak ISL = 0.5 mA −90 dBc/Hz −87 dBc/Hz −101 dBc/Hz −101 dBc/HzStrong ISL = 3 mA −98 dBc/Hz −97 dBc/Hz −92 dBc/Hz −96 dBc/Hz

ISL = 500 A (weak coupling) or ISL = 3 mA (strong coupling). The supplyvoltage is 1.8 V.

The die photo of the test circuit with the two oscillators is shown in Fig. 7.13.The total die area is 1 mm × 2 mm; this large area is necessary to allow the use ofwafer probes. The die area without pads and metal fills is 0.012 mm2 for the RCoscillator and 0.0945 mm2 for the LC oscillator. The area of the LC oscillator is 7.7times higher than that of the RC oscillator.

Strong coupling ensures accurate quadrature relationship for both oscillators.Yet, for the RC oscillator, strong coupling improves both the quadrature relationshipand the phase-noise performance (Tables 7.6 and 7.7). In the LC oscillator, strongcoupling reduces the quadrature error, but increases the oscillator phase-noise. Thishas been predicted in [20, 63] and is confirmed experimentally here.

The outputs of the oscillators were measured using G-S-G probes. Losses in thecoaxial cables and DC blocks connecting the on-wafer probes to the scope werearound 7 dB at 5 GHz. The measured output amplitude of each oscillator was closeto 100 mV. The quadrature accuracy (Table 7.7) was measured using the startingedge detection capability of the HP54120B digitizing oscilloscope.

Figures 7.14 and 7.15 show the output waveforms of the two quadrature oscilla-tors.

We can compare these oscillators using the conventional figure-of-merit [107](equation (7.7)). We also use a recently proposed figure of merit FOMA [112] thatincludes the die area:

FOMA = Lmeasured + 10 log

((Δ f

f

)2 PDC

Pref

Are f

Achip

)

(7.8)

where Achip is the circuit area in mm2, and Aref is a reference area (1 mm2).Table 7.8 shows that, with weak coupling, the LC oscillator is better than the

RC oscillator in terms of both FOM and FOMA. Yet, if one uses strong couplingto reduce the quadrature error, the noise performance of the RC oscillator improves

Table 7.7 Quadrature error of RC and LC oscillators

Coupling RC LC

Weak ISL = 0.5 mA 2.7◦ 3.5◦

Strong ISL = 3 mA 1◦ 1.5◦


Fig. 7.14 RC-oscillator quadrature outputs

Fig. 7.15 RC-oscillator quadrature outputs

7.6 Conclusions 135

Table 7.8 FOM and FOMA for the RC and LC oscillators

Coupling RC LC

FOM FOMA FOM FOMA

Weak ISL = 0.5 mA −145 dBc/Hz −164 dBc/Hz −168 dBc/Hz −178 dBc/HzStrong ISL = 3 mA −154 dBc/Hz −173 dBc/Hz −159 dBc/Hz −169 dBc/Hz

significantly, and that of the LC- oscillator is degraded. The FOM value improves,but it remains worse for the RC oscillator. The FOMA value, however, is better forthe RC oscillator with strong coupling.

It should be noted that the noise value in the LC oscillator is strongly dependenton the inductors’ Q and circuit topology. Better values may be obtained for LCoscillators different from those used here. Since in our example the simplest circuitshave been chosen for both the LC and RC oscillators, the comparison presented hereis believed to be reasonably fair.

7.6 Conclusions

In this chapter we confirm by measurements on different quadrature relaxation oscil-lators that the coupling block is critical concerning quadrature error and phase-noise.

The first prototype is an RC relaxation oscillator. In these oscillators the couplinggain should be high to force oscillators to be in precise quadrature (mismatchesand other disturbances will have then only a second order effect). Moreover, by in-creasing the coupling gain, the oscillator will improve the phase-noise performance.However, we cannot increase this gain indefinitely, because this will increase thepower dissipation and cause some reduction in the oscillation frequency.

In a second prototype we show that in quadrature LC oscillators there is a tradeoff: the coupling gain should be high to force a precise quadrature, but increasingthe coupling gain will increase the phase-noise.

The third prototype is used to demonstrate the oscillator-mixer concept, using anRC relaxation oscillator. The measurements confirm that this class of oscillators canperform wideband mixing. This oscillator-mixer circuit is well suited to be used aspart of a double-conversion receiver with a low intermediate frequency, where veryaccurate quadrature signals are necessary to reject the image signal. Measurementsshow an error of the order of 1◦ for four different die samples of the oscillator-mixer.The phase-noise measured is comparable to that of state-of-the-art RC oscillators.

The last two prototypes, at 5 GHz, show that in RC oscillators the coupling re-duces both the phase-noise and quadrature error, whereas in LC oscillators the cou-pling reduces the quadrature error, but increases the phase-noise. Figure of meritvalues indicate that quadrature RC oscillators may be a viable alternative to LCoscillators when area and cost are to be minimized.

Chapter 8Conclusions and Future Research

Contents

8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1378.2 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

8.1 Conclusions

In recent years, wireless communications have been developed due to the hugedemand for mobile equipments. Associated with the mobility, equipment low sizeand cost are important requirements. Therefore, avoiding the use of discrete com-ponents in modern transceivers is an important topic of research. Full integration ofmodern transceivers requires very accurate quadrature signals.

There are two basic receiver front-end architectures: heterodyne, which uses oneor more intermediate frequencies, and homodyne, without intermediate frequency.The main drawback of heterodyne receivers is that the image frequency band mustbe removed with external filters, which does not allow full integration. The homo-dyne approach allows full integration, since it does not have the image problem, butthe direct conversion to the baseband imposes severe restrictions (1/f noise, DC off-sets, intermodulation distortion, etc.), which are not present or can be easily avoidedin the heterodyne approach.

Since both conventional approaches have advantages and drawbacks, why notuse an approach that uses the best features the two? This is what does the low-IFreceiver. This is an heterodyne receiver, but it does not use filters to remove theimage frequency band: an image reject technique is used instead that allows the useof a very low intermediate frequency. Since external filters are not used, full integra-tion of the receiver on a single chip is possible; since there is no conversion to thebase-band, the problems of homodyne receivers are avoided. The main issue of thisapproach is that it requires very accurate quadrature outputs from the local oscillatorto cancel the image. The amount of image rejection has a direct relationship with thequadrature error. This justifies the development of new techniques to provide veryaccurate quadrature outputs.


137

138 8 Conclusions and Future Research

In this book we study and compare three types of quadrature oscillators: cross-coupled RC and LC oscillators, and the two-integrator oscillator (that has inherentlyquadrature outputs).

These three types of oscillators, despite being conceptually different, have onecommon feature: they are closed-loop structures (conventional architectures areopen-loop), which leads to accurate quadrature.

RC oscillators are known for their poor phase-noise performance when comparedwith LC oscillators, reason why they have been neglected. Although this is true fora single oscillator, it is not for cross-coupled oscillators.

In this book we study in detail the quadrature relaxation oscillators in termsof their key parameters: oscillation frequency, amplitude, quadrature relationship,and phase-noise. The effect of mismatches and other disturbances is attenuated bythe soft limiter gain, becoming a second order effect, and allowing a very accuratequadrature. We derive equations for the oscillator amplitude, which show that in thepresence o mismatches the amplitudes change to preserve the quadrature accuracy.We point out that the coupling reduces the phase-noise. In relaxation oscillatorsboth the quadrature error and the phase-noise are reduced due to the coupling. Theinfluence of coupling in the oscillator performance was confirmed by measurementson a 2.4 GHz quadrature relaxation oscillator.

Single LC oscillators are widely used in applications in which very low phase-noise is necessary. They are now the first choice in the design of modern transceivers.In this book we consider the coupling of these oscillators to obtain quadrature out-puts. We calculate the oscillation frequency of two coupled oscillators, and showthat a frequency change due to the coupling affects the Q-factor, and, therefore,degrades the phase-noise.

Comparing RC and LC quadrature oscillators, we conclude that coupling in re-laxation oscillators improves simultaneously the quadrature accuracy and phase-noise, whereas in LC oscillators the coupling improves the quadrature, but increasesthe phase-noise. Thus, in a quadrature LC oscillator there is a compromise in choos-ing the strength of coupling, which does not exist in the quadrature RC oscillator.Thus, quadrature RC oscillators can be an alternative to quadrature LC oscillators.The degradation of phase-noise due to coupling was confirmed by measurements ona 1 GHz quadrature LC oscillator.

The advantages of quadrature relaxation oscillators are left out of the conven-tional FOM (figure of merit), which considers only phase-noise and power con-sumption. There, is however, another figure of merit, FOMA, that takes also thearea into account. We have used both FOM and FOMA to compare experimentallyan RC and an LC cross-coupled oscillator at 5 GHz. RC oscillators have lowerarea, since they do not require inductors that need a large silicon area, and thisadvantage is evidenced by the FOMA values. RC oscillators can be built with a lowcost technology (they do not need special RF options, several metal layers, and thicktop metal layer, in order to improve the inductor quality factor). The conclusion ofour experimental comparison is that quadrature RC oscillators can be a suitablealternative to RC oscillators when area and cost are to be minimized.

8.2 Future Research 139

The tuning range of the previous cross-coupled oscillators (RC and LC) istypically lower than 20%, which is an important limitation, since nowadays applica-tions with wide frequency band are being developed. The two-integrator oscillatoris a single oscillator with quadrature outputs (it is not formed by two coupled os-cillators). Its phase-noise is comparable to that of a relaxation oscillator. The mainadvantage of this oscillator is the wide tuning range (close to one decade), whichmakes it a good solution for wideband or multi-standard applications.

In this book we show that the mixing function can be incorporated in the threetypes of quadrature oscillators, and that the quadrature relationship is preserved.This approach avoids separate mixers, whose mismatches are responsible for a firstorder error in the quadrature relationship, and has the advantage of saving areaand reducing the power consumption of the complete transceiver. We demonstratethe oscillator-mixer concept with measurements on a 5 GHz quadrature relaxationoscillator-mixer prototype.

8.2 Future Research

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

� Quadrature oscillators are key blocks in the design of transmitters and receivers.A natural continuation of the work in this book is to include the oscillatorsin complete receivers (especially low-IF or direct conversion receivers) andevaluate the improvement in system performance. The oscillators should alsobe included in transmitters to compare their advantages with the conventionalapproaches.

� In this book we show that the usual coupling of LC oscillators, using the firstharmonic of the signal, degrades the oscillator phase-noise. An important topic ofresearch is the development of new coupling methods (such as second harmoniccoupling) to eliminate this drawback. This work was already started, and has ledto some publications [23, 24].

� The performance of coupled LC oscillators is non-linear, but, traditionally, linearmodels are used to analyse the oscillators. A new model is required in which theoscillator is treated as a non-linear circuit. A possible approach is to consider theold non-linear differential equations of the van der Pol oscillator. The study ofthis approach has already been started [23–25]. A complete study of the stabilityof oscillations is required to find which of the different mathematically possibleoscillation frequencies are applicable.

� The main purpose of this book was the study of quadrature oscillators; however,we also started the study of the oscillator-mixer concept and we presented pos-sible implementations. This study was left at an initial stage. A comprehensivestudy of the mixing, at circuit level, should be done in terms of key parameters(for example, linearity, and noise figure); a comparative study with conventionalmixers to evaluate the advantages and disadvantages is also required.

140 8 Conclusions and Future Research

� In Chapter 6, the study of the two-integrator oscillator was confirmed by simula-tion only. A validation by measurements on a test chip of the wideband propertiesof this oscillator should be done.

� In recent years the reduction of power consumption and of supply voltage(required by modern sub-micron technologies) is an important area of research.An extension of the study in this book considering modern technologies, withlow voltage supply and with low power consumption is an important topic offuture research.

Appendix ATest-Circuits and Measurement Setup

Contents

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141A.2 Quadrature RC and LC Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

A.2.1 Test Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141A.2.2 Measurement Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

A.3 Quadrature Relaxation Oscillator-Mixer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144A.3.1 Test Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144A.3.2 Measurement Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

A.1 Introduction

In this appendix we describe in a more detailed way the test-circuits and the mea-surement setup for the five designed prototypes.

For the quadrature RC and LC oscillators we measure the quadrature error andphase-noise as a function of the coupling gain. The test-circuits are similar andthe measurement setup is the same. This is used to test four of the five prototypesmentioned in Chapter 7: a quadrature RC oscillator at 2.4 GHz, a quadrature LCoscillator at 1 GHz and two quadrature RC and LC oscillators at 5 GHz.

The remaining prototype is the oscillator-mixer at 5 GHz. We include it in aPLL and combine this with an LNA and low-pass filters, for the validation of theoscillator-mixer concept, as explained below.

A.2 Quadrature RC and LC Oscillators

A.2.1 Test Circuit

In order to have more control of the circuit variables we have an external biasingcircuit. We are able to control the tail current of the coupling differential-pairs (tochange the coupling gain), the oscillator bias current (to tune the oscillator), andthe buffers bias current (to control the output matching). For this purpose we use,respectively, the three input terminals, “coupling”, “VCO”, and “buffer”, marked inFig. A.1.

141

142 Appendix A

Fig. A.1 Test-circuit of thecross-coupled RC and LCoscillators

V

+

V

−

V

+

V

−

QuadratureOscilltor

Coupling VCO

I

Q

Buffer

Buffer

Buffer

Buffer

outI

outI

outQ

outQ

VCC

BI

M1 M1

M2 M2

M1M1M1

M2M2Vin1 Vin2 Vin3 Vin4

Vout1 Vout2 Vout3 Vout4

Fig. A.2 Output buffers

With the test-circuit represented in Fig. A.1 we are able to measure the phase-noise and quadrature error as a function of the coupling current.

The buffers at the oscillator output are needed to drive the 50 � loads. The buffersare common-drain (source-follower) stages, represented in Fig. A.2. The buffersare followed by 50 � microstrip lines on the test printed circuit board (PCB), andexternal DC blocking capacitors are used. When the buffers are biased with IB =2 mA, the oscillator output voltage is about 150 mV. The buffer implementation hasM1 transistors with (W/L) = 20 m × 1 m and M2 transistors with (W/L) =30 m × 0.35 m.

A.2.2 Measurement Setup

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

Appendix A 143

CH1

CH2

GND BufferVCO

Digital oscilloscope

CHIP

VoltageRegulator

6V 3VBatteries

Trigger

CCV

CCV

Coup

varR varRvarR Ω50

I–

I+

Q–

Q+

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

The oscillator has four outputs, but the oscilloscope has only two inputs. A prob-lem is to trigger the oscilloscope, because we use a sampling oscilloscope. Theseinstruments do not have the ability to synchronize using the input signals. Clockrecovery modules are needed if we do not have access to a trigger source. Since inour circuit we have 4 outputs, we measure two outputs with the oscilloscope, useanother output as trigger, and connect the last output to a 50 � load. It was foundthat it is very important to have the four outputs connected to 50 � loads to haveaccurate quadrature.

To prevent any interference from entering the system through the power supply,a battery supply was used. We use 4 batteries of 1.5 V and the resulting 6 V voltageis applied to a voltage regulator to produce the desired 3 V supply.

To connect the oscillator outputs to the oscilloscope we use 50 � microstriplines on the board, SMA connectors, and 50 � cables between the board and theoscilloscope.

A photo of the measurement board and batteries is shown in Fig. A.4.The circuit uses only a few external components for biasing and DC blocking.

We have used four 1 nF SMD capacitors for the four buffers DC blocking. For thepower supply decoupling we use 2 capacitors: a high frequency bypass capacitor of100 pF and a low frequency bypass capacitor of 0.47 F.

The following measurement equipment is used:

� Oscilloscope – Agilent 83484A communications analyser (50 GHz bandwidth).This is used to observe the two oscillator outputs in quadrature.

� Spectrum analyzer – Tektronix real time spectrum analyzer RSA2208A (8 GHzbandwidth). This is used to observe the output spectrum of the oscillator andmeasure the phase-noise.

144 Appendix A

Fig. A.4 Measurement setup

The organization of the test circuit (Fig. A.1) and the measurement setup (Fig. A.3)are the same for the quadrature RC and LC oscillators. The only difference is thatthe 1 GHz quadrature LC oscillator uses external inductors.

The measurements results obtained with this setup have been presented inChapter 7.

A.3 Quadrature Relaxation Oscillator-Mixer

A.3.1 Test Circuit

The PLL, LNA, and buffer are designed to support the testing of the oscillator-mixer. These circuits are well known and their design does not present major prob-lems.

The proposed quadrature oscillator-mixer is used in downconversion as shownin Fig. A.5. An RF signal at 5.01 GHz is converted to a low IF (10 MHz). After theoscillator-mixer, the I and Q outputs are filtered (for channel selection) and buffered,so that off chip loads can be driven. To drive the oscillator-mixer, off-chip signalshave to be applied to the chip. This can be done by using a low-noise amplifier(LNA) with 50 � input impedance and two current outputs. These two currents arethe bias currents of the two soft-limiters.

Relaxation oscillators are known to be noisy. To reduce the phase-noise, and toprevent frequency pulling by the input signal, the oscillator-mixer is inserted in aPLL. To keep the test circuit as simple as possible, the phase-detector and the loopfilter will be realized externally. To prevent cross-talk from a strong in-band signalto the input of the LNA, the oscillator output signal has its frequency divided by twoinside the chip.

Appendix A 145

LNA

LPF

LPFIF-I

IF-Q

RF Input

Buffer

Buffer

OscillatorMixer

Q

I

Buffer

DummyBuffer

Dividerby 2

LoopFilter

ReferenceGenerator

Test-Chip

PhaseDetector

VCO

BiasCircuit

Buffer

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

Since the oscillator-mixer is easily tuned, it is also sensitive to noise. For this rea-son, an on-chip bias block is used, which generates bias currents for all the circuitson the chip.

In this section we present the circuit schematics of the blocks in Fig. A.5 (exceptthe oscillator-mixer). All the circuits operate from a 2.5 V supply voltage.

A.3.1.1 LNA

The purpose of the LNA is to convert an input signal into the tail currents of thelimiters in the oscillator/mixer circuit. The LNA is as simple as possible, since themain purpose of the design is to show the possibilities of the oscillator-mixer.

The LO frequency is different from the RF frequency. Receiver desensitizationand DC offset due to LO self-reception are not a problem here, as they would be indirect conversion front-ends [113]. Therefore, it is not necessary to use a fully dif-ferential LNA. A single-ended implementation has lower consumption and a lowernoise figure [1]. An off-chip single-ended to differential conversion by means of abalun is not required, since the LNA has a single-ended input.

146 Appendix A

VCC

R

1Q

VinLS Cin

R

Q2

R3

Q3

IBIAS

1SLOsc 2SLOsc

LE

CE

LE

CE

Cfilt

R1 R2

Qa

Fig. A.6 LNA

The LNA schematic is shown in Fig. A.6 [1]. It consists of two inductively de-generated common emitter (CE) stages (Q transistors with emitter area 0.32 m ×10 m). Connected to the bases, is an inductor Ls (1.8 nH), which lowers the fre-quency at which the LNA input impedance is real. It is implemented using a pieceof microstrip on the printed circuit board on which the chip is mounted. This typeof LNA has reasonable noise and intermodulation performance, and offers a simplemeans to achieve impedance matching at the input. The input impedance is 50 � foreasy interface with the measuring equipment.

The frequency at which the input impedance is real, is approximately

�20 = 1

(12 Le + Ls

)

C�

(A.1)

where C� is the small-signal base-emitter capacitance of the transistors.One problem with this type of LNA is its biasing. Reasonably matched tran-

sistors, inductors, and collector currents are required. Accurate collector currentmatching is achieved here by means of resistive emitter degeneration at DC. Thedegeneration is bypassed at the frequencies of interest by using a 10 pF capacitorCE in series with L E . The bias current is obtained by mirroring a lower current.Transistor Q3 with emitter area 0.32 m × 2 m is the input of the current mir-ror, and Qa (emitter area 0.32 m × 2 m) reduces the inaccuracy due to the base

Appendix A 147

currents. All resistors that need to be matched are composed of series and parallelconnections of unit-size resistors.

The reason for the presence of the resistors R1 and R2 and the choice of theirvalues will now be explained. With IB I AS = 400 A, we wish Q1 and Q2 with5 × 400 A = 2 mA. This can be accomplished by choosing Q1 and Q2 withan emitter area 5 times larger than that of Q3. To improve the matching betweenthe emitter currents, some resistive emitter degeneration has been included, and thevoltage over the emitter resistors, R = 50 � and R3 = 250 �) is 0.1 V. To avoidinaccuracy due to the base currents, Qa is added. The impedance that is seen atthe base of the transistors of the current mirror is very low now, due to the localfeedback around Qa and Q3. To increase it to an acceptable value, R1 = 1 k�is included. This value is much higher than 50 �, the intended input impedance.However, the voltage drop over R1, due to the base currents of Q1 and Q2, wouldlead to a mismatch of the collector currents of Q1 and Q

2with respect to that of Q3.

Assuming a current gain of 100, the base current is 20 A and the voltage drop overR1 is 40 mV. To compensate this, a resistor R2 = 10 k� is inserted in series withthe base of Q3, which has the same voltage drop (namely 400 A/100 × 10 k�) asthat over R1. We also add a capacitor C f ilt (10 pF) to reduce the noise of the biasnetwork.

Simulation of the LNA has been performed to obtain its noise, distortion, gain,and input impedance. ESD input protection diodes and pad models (C4 ball)have been incorporated in the simulation. The influence of the DC decouplingcapacitor (Cin) at the bases of the transistors turned out to decrease the inputimpedance of the LNA. This was solved by increasing the value of the induc-tances in the emitter to a final value of L E = 0.72 nH (Fig. A.6). The noise fig-ure is about 2.5 dB, and the transcondutance is about 25 mA/V. The simulated IIP3

is +7 dBm.

A.3.1.2 Frequency Divider

The frequency divider divides the oscillator frequency by 2, so that it is in a differentfrequency band than the front-end input signal, thus avoiding interferences. Thefrequency divider is implemented with two current-mode logic D-type flip-flops. Adetailed explanation of the circuit performance can be found in [1].

The bias currents have been chosen such that the total divider current is low withrespect to that of the oscillator-mixer, which minimizes the overhead power con-sumption. The voltage swing in the resistors is about 300 mVpp, to ensure reliableswitching of the differential pairs in the latches.

The schematic of the frequency divider is shown in Fig. A.7 [1]. The emitter areaof the transistors is 0.32 m × 2.5 m for Q and Q1, 0.32 m × 10 m for Q2,0.32 m × 20 m for Q3, and 0.32 m × 5 m for Q4, R1 = 500 �, R2 = 125 �,R3 = 62.5 �, RC = 250 �, and C = 2 pF.

The total bias current is 3.4 mA, and simulations showed reliable operation up to10 GHz with an amplitude of 150 mV.

148 Appendix A

C

IBIAS

V −

Q Q

VCC

Q Q Q Q

Q Q

Q Q Q

Q4

CRCRRC

Q1 Q2 Q2

R2 R3 R3

Q3 Q3

Q4

R1 R2

RC

Vout

Q

in1 V −in1V

+in1V

+in1

Fig. A.7 Frequency divider

A.3.1.3 Bias Current Generator

The bias circuit generates currents IB I AS for the circuits in Figs. A.6–A.10. Thesecurrents are independent from of the supply voltage, to reduce noise couplingthrough the power supply. There are no connections off-chip to sensitive parts of thebias current generator, which further reduces sensitivity to noise. The bias currentgenerator schematic is shown in Fig. A.8 [114].

Fig. A.8 Bias currentgenerator

Qref 1

R

Qref 2

Q

Q

Q

M M M

I

Rstart

R1 R1

C

R1

mI

VCC

m sections in parallel

Appendix A 149

Fig. A.9 Divider buffer

C

BIASI

VCC

Q1

R1

V + V

–

Vout1

R

Q

Q Qin1 in1

It can be shown that current I is [114]

I = VT

Rln(n) (A.2)

where VT = kT/q, k is Boltzmann’s constant, T is the temperature in Kelvin, q is theelectron charge, and n is the emitter-area ratio of the transistors Qref 2 and Qref 1. Inthis design, Qref 1 has an emitter area of 0.8 m×10 m, and Qref 2 is composed of8 of these transistors in parallel. With the parameters chosen, I = 100 A at 20◦C.

Current I is PTAT (proportional-to-absolute-temperature) if resistor R istemperature-independent. Here, a resistor has been used which has a positive tem-perature coefficient, so the current is approximately constant: over the temperaturerange 0–70◦C the deviation from the nominal value is about ±7%.

CR1

IBIAS

VCC

Vin 1 Vin 2 Vin 3 Vin 4

Vout 3Vout 2Vout 1 Vout 4

R

Q

R

Q

R

Q

R

Q

QQQQ

Q1

Fig. A.10 Oscillator-mixer buffer

150 Appendix A

To keep the dependence of the current on process variations as small as possible,R is a wide resistor (6 m). The 3-sigma variation of the resistance value is 10%. Incombination with variation due to temperature, the reference current can deviate upto 15% from the nominal value, which was found acceptable.

A start-up circuit, composed of Q1–Q3 and Rstart with a value of 100 k� hasbeen included. The start-up current is a few nA, which is small enough not to com-promise accuracy.

To reduce the influence of shot noise and to improve matching, source degener-ation resistors are used in the current mirrors. Multiples m of the minimum currentare generated by putting m unit current sources in parallel, as shown in Fig. A.8.

The noise at high frequencies is reduced by a capacitor of 30 pF (Fig. A.8). Toachieve a high current source output impedance, long transistors (2 m) were used.Each PMOS is composed of 5 sections, 10 m wide. The values of resistors areR = 536 � and R1 = 1 k�.

A.3.1.4 Buffers

The frequency divider output buffer is represented in Fig. A.9. This buffer con-verts the divider differential output into a single-ended output: the output current atterminal Vout1 is converted to voltage with an external resistor of 50 � connectedto VCC ; Vout1 is connected to the phase-detector. The currents were dimensionedaccording to the load and the peak-to-peak output voltage of the divider, whichis about 300 mV. The implementation uses Q and Q1 transistors with emitter area0.32 m × 16.8 m and 0.32 m × 4.2 m, respectively, R = 50 �, R1 = 200 �,and C = 2 pF.

The oscillator-mixer output buffers are common-collector stages. The set of fourbuffers has the circuit in Fig. A.10. Transistors Q and Q1 emitter areas are 0.32 m×8.6 m and 0.32 m×4.3 m, respectively, R = 50 �, R1 = 100 �, and C = 2 pF.These buffers are followed by low-pass filters.

The low-pass filters (Fig. A.11) have 5th order Butterworth response with cutofffrequency 1 GHz for 3 dB attenuation. The component values are L1 = 12.221 nH,L2 = 11.04 nH, L3 = 2.472 nH, C1 = 5.39 pF, and C2 = 2.85 pF.

For good matching these filters are on-chip. In a practical implementation thesefilters can be substituted by much smaller active filters, but in this design the mainconcern is to measure the quadrature error, so we have a passive low-noise and linearfilter.

Vin Vout

L1 L1 L3

C1 C2

Fig. A.11 Low-pass filter

Appendix A 151

A.3.1.5 External blocks

The remaining external blocks to build the Phase-lock-loop (PLL) are the phase-detector and the loop filter. These blocks are connected externally. This decisionwas made to concentrate efforts on the design of the oscillator-mixer, due to timelimitations.

The phase-detector is a commercial mixer (Miteq DMX0418L), which can beused up to 18 GHz. This is followed by a first-order filter with an input impedanceof 50 �, that has one pole and one zero. Since the oscillator frequency is controlledby a current, the filter is followed by a simple transconductance amplifier with atransconductance of about 1 mA/V. A signal-independent current, for tuning theoscillator to its nominal value, can be set by means of a potentiometer.

A.3.1.6 Layout

The layout had to satisfy an important constraint imposed by the technology: the C4bondpads were not available in the fabrication run used, and had to be replaced bywirebond pads, resulting in additional inductances and in changes in the layout.

A.3.2 Measurement Setup

The measurement setup is shown in Fig. A.12. The transformers convert the bal-anced output signals from the chip to single-ended signals, suitable for measurementby an oscilloscope.

During first measurements, it was found that the output signals were disturbedby electromagnetic interference. Therefore it was decided to shield the chip andthe PLL loop filter. These were placed inside an aluminum box, and signals weretaken in and out of the box by BNC and SMA connectors (Fig. A.13). The power-on

2.5V

CH1

CH2

50W

Loopfilter

5V

Batterysupply

2.5V5V

VCC

GND

LNAIN

DIVOUT

VCOIN

Inside MetalBOX

PhaseDetector

RFinput

PLL Referenceoscillator

Digitaloscilloscope

CHIPI+

I–

Q+

Q–

Fig. A.12 Setup for measurement of quadrature error

152 Appendix A

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

switch and a variable resistor for tuning the oscillation frequency are placed outsidethe main metal box, but they are shielded inside a tin plate box with a lid. To avoidany interference from entering the system through the power supply, a battery supplywas used.

The circuit uses only a few external components for biasing and for DC block-ing of the oscillator outputs. We have used a DC blocking 800 pF SMD capacitor.Single-ended inputs and outputs have been used for RF signals, to avoid differentialto single-ended conversion on the board (for connection to measuring equipment).Only the low-IF is led off the chip differentially, since differential active measure-ment probes are available (these are modeled, for simulation, by a 50 � resistancein parallel with a 7 pF capacitance). The supply voltage is bypassed using a 0.47 Fcapacitor (low frequencies) in parallel with a capacitor of 100 pF (high frequen-cies).

Due to the unavailability of the C4 pads for flip-chip bonding, we used wirebonding to the PCB. The RF signals are led to and from the chip using 50 � mi-crostrip lines and SMA connectors. The low-IF outputs have lower frequency andare connected to transformers for conversion from differential to single-ended.

The following measuring equipment is used:

Signal Generator – HP ESG – D4000A (250 kHz–4 GHz).This is used to generate the PLL reference signal applied directly to themixer.

Appendix A 153

Signal Generator – HP 8665B (0.1 MHz–6 GHz)This is used to generate the RF signal.

Amplifier – HP 8437A (100 kHz–3 GHz)This is used to amplify and isolate the signal of the divider by 2.

Oscilloscope – Fluke PM 3380A.This is used to observe the PLL output and check if the PLL is locked.

Oscilloscope – Agilent Mixed Signal Oscilloscope 5462 2D (100 MHz–200 Ms/s)This is used to observe the two IF outputs.

Spectrum analyzer – HP 8566B (100 Hz–2.5 GHz or 2 GHz–22 GHz)This is used to observe the divider by two output.

Spectrum analyzer – HP 8568B (100 Hz–1.5 GHz)This is used to observe the IF output.

The measurements results for this oscillator-mixer have been presented in Chapter 7.

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Index

AAluminum box, 152Amplitude modulation, 77Amplitude noise, 19, 84

BBarkhausen criterion, 17BiCMOS technology, 129

CCMOS tecnhology, 121, 132

DDuty-cycle, 72

EExcess noise factor, 22

FFrequency division, 33–34

GGSM, 93

HHartley architecture, 12Havens’ technique, 34–35Heterodyne receiver, 1, 8Homodyne receiver, 2, 10

IImage rejection, 9, 13, 122Inductor

bondwire, 125external, 125integrated, 93, 132RF options, 93

Intermodulation, 27

JJitter, 19

L

(phase-noise), 19LC oscillator, see Oscillator, LCLC tank, 83Leeson’s formula, 19LNA, 145Low-IF receiver, 2, 11Figure of merit

FOM, 131, 133FOMA, 133

M

Mixers, 26Modulation, see Amplitude modulation

N

Noisefactor, 27jitter, 19phase-noise, 18transfer function, 107

O

OscillatorLC, 25figure-of-merit, 131quasi-linear oscillator, 24RC (relaxation), 24, 39strongly non-linear, 24two-integrator, 105

Oscillator-mixerLC, 96RC, 95two-integrator, 113

161

L

162 Index

PPhase difference, 73Phase-error

quadrature LC oscillator, 91quadrature RC oscillator, 138two-integrator oscillator, 111

Phase-noise1/ f noise, 11definition, 18quadrature LC oscillator, 92–96quadrature RC oscillator, 60single LC oscillator, 85single RC oscillator, 60two-integrator oscillator, 107–111

PLL, 144bias current, 148filters, 150frequency divider, 147LNA, 145–146phase-detector, 151phase-noise, 131

Q

Quadrature accuracy, see Phase-errorQuadrature LC oscillator

implementation, 85measurement setup, 119phase-error, 91phase-noise, 95Q degradation, 94test-circuit, 131

Quadrature oscillator-mixerhigh level, 64implementation, 75measurement setup, 151oscillation frequency, 72phase-error, 74phase-noise, 79test-circuit, 131

Quadrature RC oscillatorhigh level, 41–42implementation, 44measurement setup, 142–144phase-error, 41, 46phase-noise, 60–61test-circuit, 131, 142

Quality factor, Qcoupled LC oscillators, 93definition, 19two-integrator, 109

RRC-CR network, 31RC relaxation oscillator

high level, 38implementation, 39

Resonance frequency, 25, 84

SSingle LC oscillator, 82Single RC oscillator, 56

TTransmitter, 15Two integrator oscillator

implementation, 104–107non-linear, 100oscillator-mixer, 113phase-error, 111phase-noise, 110, 112quasi-linear, 102tuning range, 113

WWeaver architecture, 14White noise, 22

ZZero-IF receiver, see Homodyne receiver

Analysis and design of quadrature oscillators

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