Dynamic Translinear and Log-Domain Circuits: Analysis and Synthesis

277 Pages·1999·6.546 MB·English

by  Jan Mulder, Wouter A. Serdijn, Albert C. van der Woerd, Arthur H. M. van Roermund (auth.)

DYNAMIC TRANSLINEAR AND
LOG-DOMAIN CIRCUITS
Analysis and Synthesis
THE KLUWER INTERNATIONAL SERIES
IN ENGINEERING AND COMPUTER SCIENCE
ANALOG CIRCUITS AND SIGNAL PROCESSING
Consulting Editor: Mohammed Ismail. Ohio State University
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DYNAMIC TRANSLINEAR AND
LOG-DOMAIN CIRCUITS
Analysis and Synthesis
by
JanMulder
Philips Research Laboratories
The Netherlands
Wouter A. Serdijn
Albert C. van der Woerd
Arthur H. M. van Roermund
Delft University of Technology
The Netherlands
SPRINGER SCIENCE+BUSINESS MEDIA, LLC
ISBN 978-1-4613-7249-3 ISBN 978-1-4615-4955-0 (eBook)
DOI 10.1007/978-1-4615-4955-0
Library of Congress Cataloging-in-Publication Data
A C.I.P. Catalogue record for this book is available
from the Library of Congress.
Copyright © 1999 Springer Science+Business Media New York
Originally published by Kluwer Academic Publishers in 1999
Softcover reprint ofthe hardcover Ist edition 1999
Ali rights reserved. No part of this publication may be reproduced, stored in a
retrieval system or transmitted in any form or by any means, mechanical, photo
copying, recording, or otherwise, without the prior written permission of the
publisher, Springer Science+Business Media, LLC
Printed on acid-free paper.
Contents
Preface ix
Acknowledgements xiii
1 Introduction 1
2 Design principles 7
2.1 A general approach to companding ......... . 7
2.2 Translinear principles based on the exponential law . 15
2.2.1 Static translinear principle ......... . 15
2.2.2 Dynamic translinear principle . . . . . . . . . 18
2.3 Voltage-translinear principles based on the square law 22
2.3.1 Static voltage-translinear principle .. 22
2.3.2 Dynamic voltage-translinear principle 24
3 Analysis of trans linear circuits 29
3.1 Analysis of static translinear circuits 29
3.2 Analysis of dynamic translinear circuits 36
3.2.1 Global current-mode analysis .. 36
3.2.2 State-space current-mode analysis 44
3.2.3 Alternative analysis methods . . . 51
3.3 Characteristics of different translinear filter classes 55
3.3.1 log-domain filters. 56
3.3.2 tanh filters 61
3.3.3 sinh filters . . . . . 65
4 Synthesis of translinear circuits 73
4.1 Overview of the synthesis method . 74
4.2 Translinear transfer functions . . . 76
4.2.1 Static transfer functions .. 76
4.2.2 Dynamic transfer functions 78
4.2.3 Dimension transformations 79
4.3 Definition of capacitance currents . 83
vi Contents
4.3.1 State-space approach. . . . . . . . . . . 83
4.3.2 Linear transformations . . . . . . . . . . 86
4.3.3 Single-state non-linear transformations . 89
4.3.4 General non-linear transformations . 90
4.4 Translinear function decomposition . . . . . . . 91
4.4.1 Non-parametric decomposition ..... 92
4.4.2 An algorithm for non-parametric decomposition. 96
4.4.3 Parametric decomposition . . . 104
4.5 Hardware implementation . . . . . . . 107
4.5.1 Topology selection and biasing 108
4.5.2 Translinear devices . . . . . . . 119
4.6 Alternative synthesis methods for dynamic translinear circuits . 132
4.6.1 Synthesis based on exponential transformations . . . 132
4.6.2 Synthesis based on component substitution . . . . . 134
4.6.3 Synthesis based on Bernoulli's differential equation 135
4.7 Class-AB operation. . . . . . . . . . . . . . . . . . . . . . 136
5 Device non-idealities 141
5.1 Base currents · . 142
5.2 Parasitic resistances 143
5.3 Body effect · .. 146
5.4 Early effect .. · . 151
5.5 Parasitic capacitances 152
5.6 Mismatch . . ... 153
6 Noise 157
6.1 Definitions of dynamic range and signal-to-noise ratio 158
6.2 Transistor noise sources . . . . . . . 159
6.2.1 Bipolar transistor ...... . 159
6.2.2 Subthreshold MOS transistor 162
6.3 Noise in non-linear circuits .... 163
6.4 Noise in static translinear circuits. 165
6.4.1 Noise analysis method 165
6.4.2 Analysis examples .... . 168
6.5 Noise in translinear filters .... . 175
6.5.1 Dynamic range considerations . 175
6.5.2 Noise analysis method 179
6.5.3 Analysis examples . . . . . . . 180
Contents vii
7 Voltage-translinear circuits 191
7.1 Square law conformance . . . . . . . . . . . . . 191
7.2 Designability . . . . . . . . . . . . . . . . . . . 192
7.3 Analysis of dynamic voltage-translinear circuits 192
7.4 Characteristics of different voltage-translinear filter classes. 193
8 Realisations 197
8.1 Subthreshold MOS translinear circuits . . . . . . . . . . . .. 197
8.1.1 Bulk current-mirror ................... 198
8.1.2 A sin x-circuit in MOS technology using the back-gate 200
8.1.3 High-swing casco de MOS current mirror. . . 203
8.2 A translinear integrator for audio filter applications. . . 205
8.2.1 Design of the integrator . . . . . . . . . . . . . . 206
8.2.2 An application example for hearing instruments. 208
8.2.3 Measurement results. . . . . . . . 209
8.3 A I-volt class-AB translinear integrator . 211
8.3.1 Block schematic of the integrator. 213
8.3.2 Design of the individual blocks . . 213
8.3.3 Measurement results . . . . . . . . 217
8.4 A dynamic translinear RMS-DC converter. 218
8.4.1 Design of the RMS-DC converter . . 219
8.4.2 Measurement results . . . . . . . . . 221
8.5 A 3.3-volt current-controlled voltage-translinear oscillator 225
8.5.1 Square law conformance. . . . . . . . . . 225
8.5.2 Design of a voltage-translinear integrator 226
8.5.3 Design of the oscillator. 230
8.5.4 Measurement results . . . . . . . . . . . . 232
9 Conclusions 235
A Additional design examples 239
A.1 A syllabic companding translinear filter 239
A.1.1 Distortionless syllabic companding 240
A.1.2 Translinear implementation .. 242
A.1.3 Simulation results ...... . 245
A.2 A harmonic mean class-AB integrator 247
A.2.1 Capacitance currents . . 249
A.2.2 Design of the integrator 250
A.3 A second-order low-pass filter 254
A.3.1 Design of the filter 254
A.3.2 Simulation results .. 256
Bibliography 259
Index 271
Preface
The introduction of the capacitance as a basic TransLinear (TL) network ele
ment significantly extends the applicability of these circuits. The resulting class
of 'Dynamic Translinear' networks, also known as 'log-domain' circuits, can
be used to implement both linear and non-linear frequency-dependent signal
processing functions. In the area of analogue continuous-time filters, which is
facing serious challenges due to ever more stringent low-voltage, low-power and
high-frequency demands, the theoretically linear transfer function offered by TL
filters provides a useful alternative for those applications that do require a large
dynamic range, but do not need a high signal-to-noise ratio. Most specifications
obtainable with bipolar-transistor TL filters are comparable to the specifications
of bipolar-transistor-only gmC filters. This applies, e.g., to the signal-to-noise
ratio, the bandwidth, the power consumption and the tunability characteris
tics. However, a significantly better dynamic range specification can be realised
owing to the theoretical (external) linearity of TL filters. The dynamic range
of TL filters can even exceed the dynamic range of opamp-MOSFET-C filters,
especially at low supply voltages. Due to the promising expectations and en
couraging results obtained thus far, research efforts have rapidly increased and
dynamic translinear circuit design has become a trend.
This book describes the structured analysis and synthesis of both Static (Le.,
conventional) TransLinear (STL) and Dynamic TransLinear (DTL) circuits. It
is shown that log-domain filters, and DTL networks in general, are closely re
lated to the conventional class of STL circuits. Having established this relation,
it follows that a current-mode point of view is the most suitable approach to the
design of DTL circuitry. The current-mode approach has the additional advan
tage that the existing theory and experience on STL circuits becomes directly
applicable to the analysis and synthesis of DTL networks.
After the general introduction of Chapter 1, Chapter 2 starts with a dis
cussion of the design principles that form the foundation of the book. In a
general context, companding networks are considered that exhibit a theoreti
cally linear frequency-dependent transfer function even though the signal path
contains non-linear processing blocks. A general model is developed, includ-
x
ing both instantaneous and syllabic companding systems. Dynamic translinear
and Dynamic Voltage-TransLinear (DVTL) filters can be considered as special
cases of externally-linear internally-non-linear companding networks. Next to
the DTL and DVTL principles, the (conventional) STL principles are reviewed.
Although synthesis is more powerful than analysis, a synthesis method must
go together with a generally applicable analysis method in the same domain.
This is a prerequisite for structured electronic design. Chapter 3 therefore deals
with the analysis of TL circuits before synthesis is considered in Chapter 4. The
analysis of STL networks has been well established and is briefly reviewed in
Chapter 3. On the contrary, analysis methods for DTL circuits have not been re
ported extensively in the literature. Large-signal analysis methods are however
of crucial importance as small-signal analyses cannot prove the externally-linear
transfer function of TL filters nor reveal the functionality of non-linear DTL net
works. The existing STL analysis method uses a current-mode approach and
Chapter 3 shows that this approach is best suited to DTL circuit analysis as well.
The capacitance currents form the key to a general large-signal analysis method.
Identifying loops of capacitors and junctions, simple current-mode expressions
for the capacitance currents can be derived, which compose a supplement to
the KCL (Kirchhoff's Current Law) equations and the TL loop equations. Two
methods have been developed for finding expressions for the capacitance cur
rents. Using a global method, the final result of the calculations is a higher-order
DE describing the network. Alternatively, a state-space method can be applied,
which yields a set of first-order DEs, a state-space description, and diminishes
the intermediate expression swell. The latter method uses fictitious transistors
to convert the capacitance voltages to (collector) currents, which are chosen to
represent the state of the circuit. In addition, Chapter 3 explores the charac
teristics of three different classes of DTL networks proposed in the literature.
These are log-domain, tanh and sinh filters.
Chapter 4 constitutes the core of this book. A structured synthesis method
ology is developed for the design of both STL and DTL networks. Basically, the
method is a generalisation and extension of the existing synthesis method for
STL circuits and beneficially exploits the high level of similarity between STL
and DTL circuits. Synthesis takes off with a normalised, i.e., dimensionless,
polynomial, rational function, nth-order root function or differential equation.
Next, dimension transformations are applied to arrive at an equation having
the proper dimensions to allow for a TL implementation. Several circuit char
acteristics can be derived from the applied transformations. The subsequent
synthesis step is required only for DTL networks: the time derivatives are im
plemented by means of capacitance currents, which are introduced by definition,
using state-space techniques. The possible definitions of the capacitance cur
rents are linked to a classification of DTL circuits. The resulting framework is
found to be more general than the classifications described in the literature. As
xi
a result of this synthesis step, both STL and DTL designs are now described by
a current-mode multivariable polynomial and the succeeding design trajectory
is roughly identical. To implement the multivariable polynomial, it has to be
mapped onto one or more TL loop equations. This process is called 'translin
ear decomposition', which can be divided into non-parametric and parametric
decomposition. Next to a description of the characteristics of both types of de
compositions, an efficient algorithm is developed for the automatic generation
of non-parametric decompositions. Once a TL decomposition has been found,
the final design stage is the hardware implementation of the TL loop equations.
This process entails numerous different synthesis options and even more possi
bilities arise from the employment of alternative exponential devices, such as
compound transistors and floating-gate MOS transistors, from operation of the
MOS transistor in the triode region, and from the application of the back-gate.
Next to the synthesis method developed in this book, several alternative meth
ods have been proposed in the literature for the design of TL filters. Hence, a
comparison is made to elucidate the differences and similarities. The chapter
is concluded by a treatment of class-AB operation, which is an important issue
closely related to synthesis.
The analysis and synthesis methods dealt with in Chapters 3 and 4 are
based on ideal exponential devices. However, in practice, device non-idealities
introduce distortion. Chapter 5 treats the second-order effects involved with
the bipolar and the subthreshold MOS transistor. Methods are described to
reduce the distortion introduced by finite current gain, parasitic resistances,
body effect, Early effect, parasitic capacitances and mismatch. Nevertheless,
in general, considerable design efforts are usually required to realise a high
performance TL circuit.
Noise is of fundamental importance in electronic circuits. Chapter 6 is con
cerned with the analysis of noise in both STL and DTL circuits. Noise analysis
is not trivial due to the non-linear nature of the devices employed. Even for
externally-linear circuits, the exponential device characteristics give rise to in
termodulation between signals and noise. The situation is further complicated
by the fact that the internal noise sources are non-stationary. For these reasons,
small-signal noise analysis methods do not suffice. Once again, it is shown that
the current-mode approach, in combination with known results from non-linear
circuit theory, facilitates an elegant solution to the challenge of large-signal noise
analysis.
Chapter 7 discusses the usefulness of VTL circuits and in particular of dy
namic VTL networks. The conclusion is that VTL circuits are interesting from
an academic point of view, however, their practical value is very limited. The
decreasing validity of the square law in modern IC processes eliminates the sole
foundation of VTL circuits. Moreover, even for a perfect square law device,
the design of VTL networks is frustrated by the mathematically awkward equa-