WIDE-BANDWIDTH HIGH DYNAMIC RANGE D/A CONVERTERS THE INTERNATIONAL SERIES IN ENGINEERING AND COMPUTER SCIENCE ANALOG CIRCUITS AND SIGNAL PROCESSING Consulting Editor: Mohammed Ismail Ohio State University Related Titles: METHODOLOGY FOR THE DIGITAL CALIBRATION OF ANALOG CIRCUITS AND SYSTEMS: WITH CASE STUDIES Pastre, Marc, Kayal, Maher Vol 870, ISBN: 1-4020-4252-3 HIGH-SPEED PHOTODIODES IN STANDARD CMOS TECHNOLOGY Radovanovic, Sasa, Annema, Anne-Johan, Nauta, Bram Vol 869, ISBN: 0-387-28591-1 LOW-POWER LOW-VOLTAGE SIGMA-DELTA MODULATORS IN NANOMETER CMOS Yao, Libin, Steyaert, Michiel, Sansen, Willy Vol 868, ISBN: 1-4020-4139-X DESIGN OF VERY HIGH-FREQUENCY MULTIRATE SWITCHED-CAPACITOR CIRCUITS U, Seng Pan, Martins, Rui Paulo, Epifânio da Franca, José Vol 867, ISBN: 0-387-26121-4 DYNAMIC CHARACTERISATION OF ANALOGUE-TO-DIGITAL CONVERTERS Dallet, Dominique; Machado da Silva, José (Eds.) Vol 860, ISBN: 0-387-25902-3 ANALOG DESIGN ESSENTIALS Sansen, Willy Vol 859, ISBN: 0-387-25746-2 DESIGN OF WIRELESS AUTONOMOUS DATALOGGER IC'S Claes and Sansen Vol 854, ISBN: 1-4020-3208-0 MATCHING PROPERTIES OF DEEP SUB-MICRON MOS TRANSISTORS Croon, Sansen, Maes Vol 851, ISBN: 0-387-24314-3 LNA-ESD CO-DESIGN FOR FULLY INTEGRATED CMOS WIRELESS RECEIVERS Leroux and Steyaert Vol 843, ISBN: 1-4020-3190-4 SYSTEMATIC MODELING AND ANALYSIS OF TELECOM FRONTENDS AND THEIR BUILDING BLOCKS Vanassche, Gielen, Sansen Vol 842, ISBN: 1-4020-3173-4 LOW-POWER DEEP SUB-MICRON CMOS LOGIC SUB-THRESHOLD CURRENT REDUCTION van der Meer, van Staveren, van Roermund Vol 841, ISBN: 1-4020-2848-2 WIDEBAND LOW NOISE AMPLIFIERS EXPLOITING THERMAL NOISE CANCELLATION Bruccoleri, Klumperink, Nauta Vol 840, ISBN: 1-4020-3187-4 CMOS PLL SYNTHESIZERS: ANALYSIS AND DESIGN Shu, Keliu, Sánchez-Sinencio, Edgar Vol 783, ISBN: 0-387-23668-6 SYSTEMATIC DESIGN OF SIGMA-DELTA ANALOG-TO-DIGITAL CONVERTERS Bajdechi and Huijsing Vol 768, ISBN: 1-4020-7945-1 OPERATIONAL AMPLIFIER SPEED AND ACCURACY IMPROVEMENT Ivanov and Filanovsky Vol 763, ISBN: 1-4020-7772-6 STATIC AND DYNAMIC PERFORMANCE LIMITATIONS FOR HIGH SPEED D/A CONVERTERS van den Bosch, Steyaert and Sansen Vol 761, ISBN: 1-4020-7761-0 DESIGN AND ANALYSIS OF HIGH EFFICIENCY LINE DRIVERS FOR Xdsl Piessens and Steyaert Vol 759, ISBN: 1-4020-7727-0 LOW POWER ANALOG CMOS FOR CARDIAC PACEMAKERS Silveira and Flandre Vol 758, ISBN: 1-4020-7719-X MIXED-SIGNAL LAYOUT GENERATION CONCEPTS Lin, van Roermund, Leenaerts Vol 751, ISBN: 1-4020-7598-7 WIDE-BANDWIDTH HIGH DYNAMIC RANGE D/A CONVERTERS by Konstantinos Doris Philips Research Laboratories, Eindhoven, The Netherlands Arthur van Roermund Eindhoven University of Technology, Eindhoven, The Netherlands and Domine Leenaerts Philips Research Laboratories, Eindhoven, The Netherlands A C.I.P Catalogue record for this book is available from the Library of Congress ISBN-10 ISBN-13 ISBN-10 ISBN-13 0-387-30415-0 (HB) 978-0-387-30415-1 (HB) 0-387-30416-9 (e-book) 978-0-387-30416-8 (e-book) Published by Springer, P.O Box 17, 3300 AA Dordrecht, The Netherlands www.springer.com Printed on acid-free paper All Rights Reserved © 2006 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work Printed in the Netherlands Contents Glossary ix Abbreviations xiii Preface xv Digital to Analog conversion concepts 1.1 Functional aspects 1.1.1 Definition of the D/A function 1.1.2 Functional specifications 1.2 Algorithmic aspects 1.3 Signal processing aspects 1.3.1 Waveforms and Line coding 1.3.2 Signal Modulation concepts 1.4 Circuit aspects 1.4.1 Architecture terminology 1.4.2 Resistive voltage division architectures 1.4.3 Capacitive voltage and charge division architectures 1.4.4 Current division based architectures 1.5 Conclusions 1 11 11 13 13 14 15 16 18 18 Framework for Analysis and Synthesis of DACs 2.1 Overview 2.2 Framework description 2.2.1 Analysis 2.2.2 Synthesis 19 19 21 21 24 Contents vi Current Steering DACs 3.1 Basic circuit 3.1.1 Partitioning and segmentation 3.1.2 Current switching network and current sources 3.1.3 Clock-data synchronization circuit 3.1.4 Auxiliary circuits 3.2 Implementations and technology impact 25 25 26 29 29 30 30 Dynamic limitations of Current Steering DACs 4.1 State of the art in dynamic linearity 4.2 Dynamic limitations of current steering DACs 4.2.1 Matching and relative amplitude precision 4.2.2 Matching and relative timing precision 4.3 Conclusions 35 35 40 41 42 44 Current Steering DAC circuit error analysis 5.1 Amplitude domain errors 5.1.1 Relative amplitude inaccuracies 5.1.2 Output resistance modulation 5.2 Time domain errors 5.2.1 Nonlinear settling and output impedance modulation 5.2.2 Asymmetrical switching 5.2.3 Modulation of switching behavior 5.2.4 Charge feedthrough and injection 5.2.5 Relative timing inaccuracies 5.2.6 Power supply bounce and substrate noise 5.2.7 Clock (timing) jitter 5.3 Conclusions 45 45 45 47 48 48 51 53 54 56 59 63 66 High-level modeling of Current Steering DACs 6.1 System modeling 6.1.1 System layers 6.1.2 System excitations and responses 6.1.3 System parameters 6.1.4 Subsystem interaction 6.1.5 System modulation 6.2 Error properties and classification 6.2.1 Error properties 6.2.2 Error classification 6.3 Functional error generation mechanisms 6.3.1 Definitions 6.3.2 Algorithmic modeling 6.3.3 Functional modeling 6.3.4 Examples 67 67 68 69 69 71 72 72 73 77 79 79 80 82 85 Contents 6.4 vii Conclusions 88 Functional modeling of timing errors 7.1 Non-uniform timing 7.1.1 The Equivalent Timing error of a transition 7.1.2 Non-uniform timing in the process of signal sampling 7.1.3 Non-uniform timing in the process of signal creation 7.2 Stochastic non-uniform timing analysis 7.2.1 Correlated non-uniform timing 7.2.2 White non-uniform timing 7.2.3 RZ and NRZ waveforms 7.3 Deterministic non-uniform timing 7.3.1 Non-linear mapping of time domains 7.3.2 Non-uniform timing in signal creation 7.4 Conclusions 89 89 89 91 92 95 95 97 100 103 103 105 106 Functional analysis of local timing errors 8.1 Local timing error analysis 8.1.1 Equivalent timing error calculation 8.1.2 Signal error calculation 8.2 High level architectural parameter tradeoffs: 8.3 Conclusions 109 109 109 113 116 118 Circuit analysis of local timing errors 9.1 Circuit analysis with linear models 9.1.1 Circuit behavioral-level analysis of timing errors in a chain 9.1.2 Transistor level analysis 9.2 Local timing error tradeoffs 9.2.1 Switch timing errors 9.2.2 Latch timing errors 9.3 Conclusions 119 119 120 126 135 135 137 137 10 Synthesis concepts for CS DACs 10.1 Information management in the CS DAC 10.1.1 The basic current steering DAC hardware 10.1.2 Information sources 10.1.3 Optional hardware: detection and control operations 10.1.4 Algorithms 10.1.5 Space/Time error mapping and processing 10.2 Synthesis Policy 10.3 A-posteriori error correction methods 10.3.1 Calibration in amplitude and time domain 10.3.2 Generalized mapping 10.3.3 Applications of generalized mapping 139 139 141 141 142 143 145 146 148 148 151 155 segmentation viii Contents 10.3.4 Realization issues of the generalized mapping concept 156 10.4 Conclusions 157 11 Design of a 12 bit 500 Msample/s DAC 11.1 Design approach 11.2 Architecture 11.2.1 Signaling and circuit logic 11.2.2 Power supply and biasing 11.2.3 Thermometer/binary bits partitioning 11.3 Switched-Current cell 11.3.1 Current source 11.3.2 Switch 11.4 Decoder, data synchronization and conditioning 11.4.1 Binary-to-Thermometer decoder 11.4.2 Delay equalization 11.4.3 Master-slave latches and drivers 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Current-Steering DACs,” in European Solid-State Circuits Conf (ESSCIRC), 2001 [177] Y Cong and R.L Geiger, “A 1.5V 14b 100MS/s Self-Calibrated DAC,” in IEEE International Solid-State Circuits Conf (ISSCC), 2003, pp 128–130 A Output spectrum for timing errors A.1 Power spectrum of y(t) for random timing errors The power spectrum of y(t) from eq (7.10) will be calculated The assumptions made are that the DAC input z(m) is a ergodic stationary random process, and the timing errors a stationary random process Then y(t) is cyclostationary Initially, the mean of the empirical autocorrelation Rˆ y (t,t + τ ) is evaluated Then we will obtain the mean of the empirical power spectrum E{Sˆy ( f )} with a Fourier transform of Rˆ y (t,t + τ ), from which the averaged probabilistic power spectrum Sy ( f ) is extracted since they are equal due to regularity [128]1 The probabilistic power spectrum Sy ( f ) matches that of Sy ( f ) due to stationarity We start with the evaluation of the expected empirical autocorrelation E{Rˆ y (τ )}: E{Rˆ y (τ )} = E{ lim D→∞ 2D D ∑ ∑ z(m + q)z(m)δ (t + τ − (q + m)Ts − µm+q )δ (t − mTs − µm )dt} −D q m D→∞ 2D = E{ lim D→∞ 2D +E{ lim D D ∑ z2 (m)δ (t + τ − mTs − µm )δ (t − mTs − µm )dt} −D m ∑ ∑ z(m + q)z(m)δ (t + τ − (q + m)Ts − µm+q )δ (t − mTs − µm )dt} −D q=0 m (A.1) Regularity guarantees the existense of time average limits such as the empirical mean and autocorrelation It is less strong than ergodicity 199 Appendix A Output spectrum for timing errors 200 Let the first and the second parts of the sum of eq (A.1) be T1 and T2 , respectively Recall the definition of the expectation of a function F(x) is E{F(x)} = σ f (σ )F(σ )d σ with f (σ ) being the pdf of the random variable x In our case f (σ ) is the probability density function of the time-jitter µm The term T1 is changed to (N+ 21 )Ts N →∞ (2N + 1)Ts T1 = E{ lim ∑ z(m)2 δ (t + τ − mTs − µm )δ (t − mTs − µm )dt} −(N+ 21 )Ts m (A.2) Since µm are in the neighborhood of mTs , we may rewrite (A.2) as N Rz (0) ∑ N →∞ (2N + 1)Ts m=−N T1 = lim ∞ −∞ E{δ (t + τ − mTs − µm )δ (t − mTs − µm )}dt = Rz (0)δ (τ ) Ts (A.3) The term T2 is written as T2 = E{ (N+ 21 )Ts 1 · lim Ts N →∞ 2N + ∑ ∑ z(m + q)z(m)δ (t + τ − (q + m)Ts − µm+q )δ (t − mTs − µm )dt} −(N+ 21 )Ts q=0 m N 1 lim z(m + q)z(m) = E{ ∑ ∑ Ts N →∞ 2N + q=0 m=−N ∞ −∞ (A.4) δ (t + τ − qTs − mTs − µm+q )δ (t − mTs − µm )dt} and if we use Im+q,m (τ ) = ∞ −∞ δ (t + τ − qTs − mTs − µm+q )δ (t − mTs − µm )dt (A.5) we transform T2 to T2 = N 1 lim E{z(m + q)z(m)}E{Im+q,m (τ )} ∑ ∑ Ts N →∞ 2N + q=0 m=−N (A.6) Next, we use the joint PDF cn−m (tn ,tm ) of the jitter to write kq (τ ) = E{Im+q,m (τ )} as +∞ kq (τ ) = δ (t + τ − (q + m)Ts − µm+q )δ (t − mTs − µm )cq (µm+q , µm ) d µm+q d µm dt −∞ (A.7) A.1 Power spectrum of y(t) for random timing errors 201 and finally kq (τ ) = ∞ −∞ cq (t + τ − qTs ,t)dt (A.8) The time averaged probabilistic autocorrelation is obtained combining the equations (A.8) (A.4) and (A.6) into E{Rˆ y (τ )} = 1 Rz (0)δ (τ ) + ∑ Rz (q)kq (τ ) Ts Ts q=0 (A.9) The function Rz (q) = E{z(m + q)z(m)} represents the probabilistic autocorrelation of the stationary input signal z(m), and Rz (0) is its power The next step is to use eq (A.9) and with a Fourier transformation to obtain the power spectrum of the process y(t) The difficulty is posed by the transformation of kq (τ ), defined as Kq ( f ) Therefore, we define the double Fourier integral of the jitter Mk−l ( fk , fl ) for k = l Mk−l ( fk , fl ) = ∞ ∞ −∞ −∞ fk−l (µk , µl )e− j2 pi( fk µk + fl µl ) (A.10) Observe that Mk−l ( fk , fl ) is related to the characteristic function Ck−l ( fk , fl ) = E{e− j2π ( fk µk + fl µl ) } Because the timing error process is assumed stationary the characteristic function Cm,n ( f , − f ) depends only on the difference k − l = q, hence C0 ( f ) = |Cq ( f )| ≤ (A.11) (A.12) Then it is easy to show that for q = Kq ( f ) = e− j2π f qTs Mq ( f , − f ) = e− j2π f qTs = E{e− j2π f (µm+q +µm ) } = e− j2π f qTs Cq ( f , − f ) (A.13) The Fourier transformation of eq (A.4) with the use of eq (A.10) gives Sy ( f ) = Rz (q)Cq ( f , − f )e− j2π q f Ts Ts ∑ q (A.14) Eq (A.14) gives us the power spectrum of the impulse position modulated waveform that is subject to stationary timing uncertainties with for general statistical properties and correlation The analysis when z(m) is deterministic is very similar In place of the probabilistic autocorrelation function Rz (q) of the stationary signal z(m) the empirical autocorrelation Rˆ z (q) is used This follows directly from eq (A.2) and (A.4) where the factors T1 and T2 are calculated Indeed, if z(m) is not a random process, the expectation in eq does not apply to the factors z2 (m) and z(m + q)(z(m), which subsequently are combined with the N discrete average operant = 2N+1 ∑m=−N to form Rˆ z (0) and Rˆ z (q), respectively Appendix A Output spectrum for timing errors 202 A.2 Spectrum of y(t) for deterministic timing errors The general magnitude spectrum will be calculated in this section The following two properties of the Bessel function of the first kind are used for the calculations: eM sin θ = ∑ Jk (M)e jkθ (A.15) k J−k (x) = (−1)k Jk (x) (A.16) (A.17) The time modulated signal z(t − µ (t)) is calculated first Using the definition of the Bessel function, it is found that z(t − µ (t)) = P (−1) ∑ ∑ A p Jq (ω p M)e j(ω p +qωµ )t p=1 q P (A.18) = (−1) ∑ ∑ B p,q (ω p M) cos((ω p + qωµ )t) p=1 q where B p,q (ω p M) = A p Jq (ω p M) Next we calculate in the same way the factor (−1) e jmωs (t −µ (t)) = ∑ ∑ Jr (mωs M)e j(mωs +rωµ )t Ts ∑ T s m m r (A.19) The combination of eq (A.18) and (A.19) gives y(t) = Ts P ∑ ∑ cos((ω p + qωµ )t)B p,q (ω p M)Jr (mωs M)e j(mωs +rωµ )t (A.20) p=1 q,m,r where we have used Γ p,q,r (ω p M, mωs M) = B p,q (ω p M)Jr (mωs M) Applying a Fourier transformation in y(t) leads to Y(f) = Ts Γ p,q,r (ω p M, mωs M) δ f − m fs − r f µ ± ( f p + q f µ ) p=1 q,m,r P ∑ ∑ (A.21) We define fA (p, q) = f p + q f µ and fB (m, r) = m fs + r f µ , and we evaluate the magnitude spectrum |Y ( f )| |Y ( f )| = Ts |Γ p,q,r (ω p M, mωs M)| [δ ( f − fB (m, r) ± fA (p, q))] p=1 q,m,r P ∑ ∑ (A.22) B Literature data 203 204 Appendix B Literature data ... MIXED-SIGNAL LAYOUT GENERATION CONCEPTS Lin, van Roermund, Leenaerts Vol 751, ISBN: 1-4020-7598-7 WIDE- BANDWIDTH HIGH DYNAMIC RANGE D/A CONVERTERS by Konstantinos Doris Philips Research Laboratories, Eindhoven,... Vol 868, ISBN: 1-4020-4139-X DESIGN OF VERY HIGH- FREQUENCY MULTIRATE SWITCHED-CAPACITOR CIRCUITS U, Seng Pan, Martins, Rui Paulo, Epifânio da Franca, José Vol 867, ISBN: 0-387-26121-4 DYNAMIC. .. ANALOGUE-TO-DIGITAL CONVERTERS Dallet, Dominique; Machado da Silva, José (Eds.) Vol 860, ISBN: 0-387-25902-3 ANALOG DESIGN ESSENTIALS Sansen, Willy Vol 859, ISBN: 0-387-25746-2 DESIGN OF WIRELESS AUTONOMOUS DATALOGGER