Analog and Telecommunication Electronics

Politecnico di Torino - ICT School Analog and Telecommunication Electronics D3 - A/D converters» Error taxonomy» ADC parameters» Structures and taxonomy» Mixed converters» Origin of errors 12/05/2011-1 ATLCE - D3-2010 DDC

Lesson D3: ADC taxonomy and errors Analog to Digital converters Transfer function and error taxonomy (linear, nonlinear, dynamic) Converter parameters: complexity and speed Examples of A/D converters Flash, Tracking, Successive approximation Residue, subranging, folding structures Pipeline structures Mixed structures Performance tradeoff Text reference Sect. 4.3 12/05/2011-2 ATLCE - D3-2010 DDC

Errors in A/D converters Dual transfer function vs D/A X axis: analog values (continuous) Y axis: numeric values (discrete) Each A Di interval corresponds to a D i value stair transfer function If N large looks as a continuous line Same error classification as D/A» Linear: offset and gain» Nonlinearity: differential and integral» Dynamic parameters 12/05/2011-3 ATLCE - D3-2010 DDC

Ideal A/D transfer function D (digital) M Di 2 1 0 1 LSB A (analog) A Di A D 0 S 12/05/2011-4 ATLCE - D3-2010 DDC

Static errors: two-steps analysis Linear approximation of actual transfer function Actual transfer function vs linear approximation Nonlinearity errors: integral nonlinearity Linear approximation vs ideal transfer function Linear errors: offset and gain Detailed quantization interval analysis Differential nonlinearity 12/05/2011-5 ATLCE - D3-2010 DDC

Actual vs ideal transfer function M D ideal 0 0 Actual transfer function S A 12/05/2011-6 ATLCE - D3-2010 DDC

Best linear approximation M D Best approx. straigth line 0 0 Actual transfer function S A 12/05/2011-7 ATLCE - D3-2010 DDC

Linear approx. vs ideal transf. function M D ideal G Best approx. straigth line O 0 0 S A 12/05/2011-8 ATLCE - D3-2010 DDC

Integral nonlinearity error M D integral nonlinearity Nonlinearity band error inl Best approx. straigth line 0 0 Actual transfer function S A 12/05/2011-9 ATLCE - D3-2010 DDC

Complete view M D integral nonlinearity Nonlinearity band ideal error inl Best approx. straigth line 0 0 Actual transfer function S A 12/05/2011-10 ATLCE - D3-2010 DDC

Linear and nonlinear errors Offset error shift of the output Gain errors rotation of the transfer function Both can be compensated with gain and offset corrections in the signal chain Integral nonlinearity error inl Position dependent; cannot be compensated 12/05/2011-11 ATLCE - D3-2010 DDC

Ideal A/D transfer function Uniform quantization: All quantization intervals AD have the same amplitude M D A D 0 0 S A 12/05/2011-12 ATLCE - D3-2010 DDC

Integral nonlinearity error Actual quantization intervals: A D A D Differential nonlinearity: DNL = A D -A D M D Ideal transfer function A D 0 A D 0 S Actual transfer function A 12/05/2011-13 ATLCE - D3-2010 DDC

Missing code error Wide intervals reduce adjacent ones Very wide intervals can suppress adjacent ones Missing code error: DNL > 1 LSB M D A D1 A D2 0 0 S This code is never generated A 12/05/2011-14 ATLCE - D3-2010 DDC

Integral and differential nonlinearity Integral nonlinearity INL How much the global transfer function deviates from a straight line Unique figure Differential nonlinearity DNL Difference A D -A D between ideal (A D ) and actual (A D ) amplitude of each quantization interval Specific to each interval (but has a max!) INL = ( DNL ) = ( DNL ) DNL = Der( INL ) 12/05/2011-15 ATLCE - D3-2010 DDC

Relation Integral/differential nonlinearity ε INL = ( DNL ) = ( DNL ) ε DNL = Der( INL ) Example Fixed polarity ε DNL high ε INL Alternate polarity ε DNL low ε INL 12/05/2011-16 ATLCE - D3-2010 DDC

Dynamic parameters Conversion from A to D requires some time: conversion time Tc Can be specified also as conversion frequency Fc = 1/Tc In most cases The ADC receives a Conversion Start (CS) command After Tc the ADC raises a End Of Conversion (EOC) flag. Some ADC can follow a (slowly) changing signal: tracking converters Dynamic behavior specified by max track rate (Slew Rate) 12/05/2011-17 ATLCE - D3-2010 DDC

ADC error summary Linear errors: Gain: G offset: O Nonlinearity errors: Integral NL: inl differential NL: dnl Dynamic parameters Conversion time: t C Tracking rate: dv/dt 12/05/2011-18 ATLCE - D3-2010 DDC

Lesson D3: A/D converters Analog to Digital converters Error taxonomy (linear, nonlinear, dynamic) Converter parameters: complexity and speed Examples of A/D converters Flash, Tracking Successive approximation, Residue/subranging Pipeline structures Performance tradeoff 12/05/2011-19 ATLCE - D3-2010 DDC

A/D converter architectures Various types of ADC, which can be classified from: Complexity» Number of comparators in the circuit» Better if few comparators Conversion time Tc» Or conversion rate Fc, inverse of Tc» Better if low Tc (high speed, high Fc) Linked parameters: High speed converters are more complex High speed and high resolution are expensive. 12/05/2011-20 ATLCE - D3-2010 DDC

A/D converters classification Complexity Conv time Parallel (flash) Pipeline Residue Successive approximation Tracking, Ramp Complexity: number of comparators. Conversion time: Tc = 1/Fc (inverse of the number of conversion/s). 12/05/2011-21 ATLCE - D3-2010 DDC

Parallel (flash) ADC V R - + 0 thermometric or linear output coding - + 0 A - + 1 - + 1 Analog input level (between 0 and V R ) 12/05/2011-22 ATLCE - D3-2010 DDC

Flash ADC with coded output V R M N encoder - 0 + A - + - + 0 1 M-N enc. - 1 + M = 2 N -1 N 12/05/2011-23 ATLCE - D3-2010 DDC

Flash converter parameters 2 N comparators (2 N -1) 1 comparison cycle for N bits Fast all comparators operate in the same time slot Complex requires many comparators 12/05/2011-24 ATLCE - D3-2010 DDC

DAC - feedback converters A logic network builds an approximation A of the input A using results of A<--> A comparison A is obtained from D through a DAC D is the numeric representation of A A A + - Logic network DAC D 12/05/2011-25 ATLCE - D3-2010 DDC

Feedback converters algorithms The logic network modifies D till A becomes the best approximation of A (within the N-bit resolution) Two procedures: Steps of one LSB: tracking converters 2 N steps for full scale change; conversion time: T C = 2 N T CK Start from MSB: successive approximation converters N steps for any conversion: T C = N T CK 12/05/2011-26 ATLCE - D3-2010 DDC

LSB steps: staircase converter Counter with enable If A < A, EN = 1 (counter enabled) If A > A, EN = 0 (counter disabled) M = 2 N steps from 0 to S:T C = 2 N T CK A A + - EN counter CK DAC D 12/05/2011-27 ATLCE - D3-2010 DDC

Signal acquisition with staircase ADC As long as A < A, EN = 1 and counter is enabled, The feedback signal A goes up 1 LSB for each Tck As A > A the comparator disables the counter; A stops max conversion time: Tc max (0 to S change of A) = 2 N Tck For new conversion reset counter to 0 12/05/2011-28 ATLCE - D3-2010 DDC

LSB steps: tracking converter Up/Down counter If A < A, D = D + 1 If A > A, D = D - 1 M = 2 N steps from 0 to S:T C = 2 N T CK A A + - U/D U/D counter CK DAC D 12/05/2011-29 ATLCE - D3-2010 DDC

Acquisition of constant signal As long as A < A, counter goes UP on CK edges The feedback signal A moves 1 LSB for each Tck As A > A the counter reverses count direction max conversion time: Tc max (0 to S change of A) = 2 N Tck 12/05/2011-30 ATLCE - D3-2010 DDC

Tracking of changing signals The converter can track signals with dv/dt < A D /Tck Overload Tracking 12/05/2011-31 ATLCE - D3-2010 DDC

Tracking converter parameters 1 comparator 2 N comparison cycle for N bits Slow fully sequential decisions limited dv/dt tracking capability Simple requires a single comparator 12/05/2011-32 ATLCE - D3-2010 DDC

Start with MSB: succ. approx. ADC Input signal compared with S/2: result --> MSB MSB = 0: next comparison with S/4 MSB = 1: next comparison with 3S/4 result --> MSB - 1.. A A + - Successive approximation register (SAR) DAC D 12/05/2011-33 ATLCE - D3-2010 DDC

Approximation sequence - 1 Input signal A S A S/2 0 t 12/05/2011-34 ATLCE - D3-2010 DDC

Approximation sequence - 2 A is compared with S/2 A = S/2 by setting MSB = 1 S A S/2 0 t A = S/2 12/05/2011-35 ATLCE - D3-2010 DDC

Approximation sequence - 3 Since A > S/2, MSB = 1 0 - S/2 excluded from possible A values S A S/2 0 t A = S/2 A > A MSB = 1 12/05/2011-36 ATLCE - D3-2010 DDC

Approximation sequence - 4 A compared with mid-value of possible range A = 3S/4 by setting MSB-1 = 1 S A 3S/4 S/2 0 t A = S/2 A > A MSB = 1 A = 3S/4 12/05/2011-37 ATLCE - D3-2010 DDC

Approximation sequence - 5 Since A < 3S/4, MSB -1 = 0 3S/4 - S range excluded from possible A values S A 3S/4 S/2 0 t A = S/2 A > A MSB = 1 A = 3S/4 A < A MSB-1 = 0 12/05/2011-38 ATLCE - D3-2010 DDC

Approximation sequence - 6 A is compared with mid-value of possible range A = 5S/8 by setting MSB - 2 = 1 S A 3S/4 S/2 0 t A = S/2 A > A MSB = 1 A = 3S/4 A < A MSB-1 = 0 A = 5S/8 12/05/2011-39 ATLCE - D3-2010 DDC

Approximation sequence - 7 Since A < 5S/8, MSB - 2 = 1 S/2-5S/8 range excluded from possible A values S A 3S/4 S/2 0 t A = S/2 A > A MSB = 1 A = 3S/4 A < A MSB-1 = 0 A = 5S/8 A > A MSB-2 = 1 A = 11S/16 12/05/2011-40 ATLCE - D3-2010 DDC

Approximation sequence - 8 Since A < 11S/16, MSB - 3 must be 0 11S/16-3S/4 range excluded from possible A values S A 3S/4 S/2 0 t A = S/2 A > A MSB = 1 A = 3S/4 A < A MSB-1 = 0 A = 5S/8 A > A MSB-2 = 1 A = 11S/16 A < A MSB-3 = 0 12/05/2011-41 ATLCE - D3-2010 DDC

Complete decision tree S 3S/4 S/2 0 t Sequence of possible output A from D/A which are compared with A 12/05/2011-42 ATLCE - D3-2010 DDC

Successive approx. ADC parameters Single comparator N comparison cycles for N-bit conversion Vs flash ADC Simpler: 1 comparator vs 2 N Slower: N steps vs 1 Vs tracking ADC Same complexity: 1 comparator faster: N steps vs 2 N 12/05/2011-43 ATLCE - D3-2010 DDC

Speed vs complexity Complex Conv time Parallel (flash) 2 N 1??? (Pipeline) N 1??? (Residue) N N Successive Approx 1 N Tracking 1 2 N Complexity: Conversion time: proportional to the number of comparators. the maximum number of comparator delay (clock periods) to complete a conversion the table suggest two new types of ADC. 12/05/2011-44 ATLCE - D3-2010 DDC

Comparison in SAR ADC Sequence of comparison in the SAR ADC: MSB: A > S/2? MSB-1: A > S/4 + S/2 MSB? A > S/4 if MSB = 0 A > S/4 + S/2 if MSB = 1 A - S/2 MSB > S/4? 2(A - S/2 MSB) > S/2? A - S/2 MSB = R1 (MSB residue) MSB-1 decision becomes 2 R1 > S/2? Similar to MSB decision (comparison A S/2) 12/05/2011-46 ATLCE - D3-2010 DDC

Comparison in SAR ADC Sequence of comparison in the SAR ADC: MSB: A > S/2? MSB-1: A > S/4 + S/2 MSB? A > S/4 if MSB = 0 A > S/4 + S/2 if MSB = 1 A - S/2 MSB > S/4? 2(A - S/2 MSB) > S/2? A - S/2 MSB = R1 (MSB residue) MSB-1 decision becomes 2 R1 > S/2? The SAR algorithm can find bit(msb-i) by comparing residue Ri with S/2 MSB: i = 0; MSB-1: i = 1;. 12/05/2011-47 ATLCE - D3-2010 DDC

Subranging/residue converter The residue (order M) is the difference between A and its M-bit approximation on each step the residue is amplified (x 2) and compared again with S/2 S A S/2 R1 2 R1 2 R2 R3 0 R2 t S = S/2 MSB = 1 S = S/2 MSB-1 = 0 S = S/2 MSB-2 = 1 12/05/2011-48 ATLCE - D3-2010 DDC

Subranging ADC procedure - 1 At first step A is compared with S/2 Since A > S/2, MSB = 1 S A S/2 0 S = S/2 A > S/2? t 12/05/2011-49 ATLCE - D3-2010 DDC

Subranging ADC procedure - 2 The MSB residue is R1 R1 is amplified (x 2) and compared with S/2 S A S/2 R1 2 R1 0 S = S/2 MSB = 1 S = S/2 2R1 > S/2? t 12/05/2011-50 ATLCE - D3-2010 DDC

Subranging ADC procedure - 3 Since 2R1 < S/2, MSB-1 = 0 The MSB-1 residue is R2 R2 is amplified (x 2)and compared with S/2 S A S/2 R1 2 R1 2 R2 0 S = S/2 MSB = 1 R2 S = S/2 MSB-1 = 0 S = S/2 2R2 > S/2? t 12/05/2011-51 ATLCE - D3-2010 DDC

Subranging ADC procedure - 4 Since 2R2 > S/2, MSB-2 = 1 The MSB-2 residue is R3 R3 is amplified (x 2)and compared with S/2 S A S/2 R1 2 R1 2 R2 R3 0 S = S/2 MSB = 1 R2 S = S/2 MSB-1 = 0 S = S/2 MSB-2 = 1 t 12/05/2011-52 ATLCE - D3-2010 DDC

Subranging ADC procedure - 5 Ri = A (i-bit approximation of A) At each step the residue Ri is amplified (x 2) and compared with S/2 S A S/2 R1 2 R1 2 R2 R3 0 S = S/2 MSB = 1 R2 S = S/2 MSB-1 = 0 S = S/2 MSB-2 = 1 t 12/05/2011-53 ATLCE - D3-2010 DDC

Residue converter parameters The residue ADC uses, for each bit One comparator (for bit value decision) One 1-bit DAC (to build the approximation) One (analog) adder (to evaluate the residue) One amplifier (gain = 2) (to bring residue to full scale) 12/05/2011-54 ATLCE - D3-2010 DDC

Block diagram of residue ADC or 12/05/2011-55 ATLCE - D3-2010 DDC

Precision in subranging converters Any error in residue evaluation is propagated to the following stages residue must be evaluated with a resolution corresponding to the residual bit number» ADC precision» DAC precision» Amplifiers and S/H (pipeline) precision Previous example First stage ADC and DAC (1-bit) need 8-bit precision precision decreases towards LSBs 12/05/2011-56 ATLCE - D3-2010 DDC

Subranging ADC parameters N comparators N comparison cycles for N-bit conversion Vs successive approximation ADC Higher complexity: N vs 1 comparator Same speed: N steps No benefit Useful as starting structure for pipeline ADC 12/05/2011-57 ATLCE - D3-2010 DDC

Multistage pipeline converters Operate on different samples at the same time Input sample 1 Input sample 2, processa 1 Input sample 3, processa 2, processb 1.. ProcessZ 1. Output sample 1 Needs analog memory elements 12/05/2011-59 ATLCE - D3-2010 DDC

Pipeline sequence Input sample sequence: A, B, C, D,... Processing stage time t1 t2 t3 t4 1 A B C D 2 X A B C 3 X X A B 4 X X X A result of sample A conversion available 12/05/2011-60 ATLCE - D3-2010 DDC

Pipeline-subranging ADC 12/05/2011-61 ATLCE - D3-2010 DDC

Comparison with other techniques A N-bit pipeline A/D converter uses: N comparators N comparison cycles (to complete the conversion of each sample) Conversion time: N-cycle latency 1-cycle conversion (throughput) Same speed as a flash with N comparators (2 N in the flash) 12/05/2011-62 ATLCE - D3-2010 DDC

Speed vs complexity Complex Conv time Latency Parallel (flash) 2 N 1 1 Pipeline N 1 N Residue N N N Successive Approx 1 N N Tracking 1 2 N 2 N Complexity: Conversion time: Latency: proportional to the number of comparators. the maximum number of comparator delay (clock periods) to complete a conversion delay to get the result. 12/05/2011-63 ATLCE - D3-2010 DDC

Classic 8-bit converters 8-bit Flash: 2 8-1 = 255 comparators; T CT = T C 8-bit SAR: 1 comparator; T CT = 8 T C + 7 T DA Limited choice No room for tradeoff Can we do something between? Faster than SAR, slower than flash Less expensive than flash (less comparators) 12/05/2011-64 ATLCE - D3-2010 DDC

Example: best cost/speed architecture Goal: 8-bit ADC Conversion time Tct < 60 ns Devices available: comparators with Tc = 10 ns Suitable design SAR: conversion time Tc = 8 x 10 = 80 ns too long Flash: Tc = 10 ns, but» Expensive: 255 comparators» Overkilling: 60 ns is enough Any other choice? 12/05/2011-65 ATLCE - D3-2010 DDC

Multibit residue architectures First-stage conversion on 1, 2, 3, M bits Comparators are replaced by M-bit ADC The approximation is built using M-bit DAC Residue has a max value Ad = S/ 2 M Gain required to bring residue up to S: 2 M Multibit-residue provides various speed/complexity tradeoffs Higher speed, with the same number of comparators Less comparators for the same speed Better tuning of design to specifications 12/05/2011-66 ATLCE - D3-2010 DDC

Choices for 8-bit converters Comparators, conversion time (T CT ), latency (T L ) 8-bit Flash: 2 8-1 = 255 comparators; T CT = T C T L = 1 8-bit SAR: 1 comparator; T CT = 8 T C + 7 T DA T L = 8 Something in the middle: multibit residue subranging converters Two cascaded 4-bit flash: 2(2 4-1) = 30 comparators; T CT = 2T C +T DA T L = 2 Four cascaded 2-bit flash: 4(2 2-1) = 12 comparators; T CT = 4T C + 3 T DA T L = 4 12/05/2011-67 ATLCE - D3-2010 DDC

Multibit residue ADC: 2 4-bit cells A/D - 4bit + 16 D/A - 4bit MSB,.. (D7, 6, 5, 4) 8-bit residue ADC: 2 4-bit cells Total conversion time: Tc = 2 x Tc(A/D) + Ta(D/A) A/D - 4bit D3, 2, 1, 0 12/05/2011-68 ATLCE - D3-2010 DDC

Multibit residue ADC: 4 2-bit cells A/D - 2bit + 4 D/A - 2bit MSB, MSB-1 (D7, D6) 8-bit residue ADC: 4 2-bit cells Total conversion time Tct = 4 x Tc(A/D) + 3 x Ta(D/A) A/D - 2bit D5, D4 + D/A - 2bit 4 A/D - 2bit D3, D2 + 4 D/A - 2bit A/D - 2bit LSB+1, LSB (D1, D0) 12/05/2011-69 ATLCE - D3-2010 DDC

Multibit residue ADC comparison A/D - 4bit + 16 D/A - 4bit MSB,.. (D7, 6, 5, 4) A/D - 2bit + 4 D/A - 2bit MSB, MSB-1 (D7, D6) A/D - 4bit D3, 2, 1, 0 A/D - 2bit D5, D4 8-bit residue ADC: 2 4-bit cells Tct = 2 x Tc(A/D) + Ta(D/A) + 4 D/A - 2bit A/D - 2bit D3, D2 + D/A - 2bit 8-bit residue ADC: 4 2-bit cells Tct = 4 x Tc(A/D) + 3 x Ta(D/A) 4 A/D - 2bit LSB+1, LSB (D1, D0) 12/05/2011-70 ATLCE - D3-2010 DDC

Speed vs complexity with multibit Complex Speed Conv time Latency Parallel (flash) 2 N 1 1 Pipeline N 1 N Residue N N N Residue K bit N/K N/K N Successive Approx 1 N N Tracking 1 2 N 2 N. 12/05/2011-71 ATLCE - D3-2010 DDC

Mixed and multistage converters Application of subranging to bit groups (K-bit) Comparator N bit ADC Residue evaluation N bit DAC + differential amplifier With K-bit groups N/K stages Interstage amplifier gain 2 N Basic ADC: FLASH Complexity: Ps = 2 K total: P T = N/K x 2 K Conversion time T C = 1 total: T CT = N/K x T C Basic ADC: SAR Complexity: Ps = 1 total: P T = N/K Conversion time T C = K total: T CT = N x T C 12/05/2011-72 ATLCE - D3-2010 DDC

Pipeline converters Pipeline can be usd for any multistage Reduces the equivalent conversion time T CT Faster equivalent conversion rate F S Same latency time Pipeline = Residue + Memory element (analog) at the input of each stage Change each amplifier into Sample/Hold + amplifier Some added complexity Multibit residue pipeline FLASH ADC: equivalent T CT = 1 (more precisely T C +T DA ) SAR ADC: equivalent T CT = K T C (more precisely K (T C +T DA )) Additional design freedom 12/05/2011-73 ATLCE - D3-2010 DDC

Speed vs complexity with multibit Complex Speed Conv time Latency Parallel (flash) 2 N 1 1 Pipeline N 1 N Residue N N N Residue K bit Flash (2 K )N/K N/K 1 Residue K bit SAR N/K N N Successive Approx 1 N N Tracking 1 2 N 2 N. 12/05/2011-74 ATLCE - D3-2010 DDC

Speed vs complexity - graph Complexity (log) 2 N Flash N Pipeline Residue 1 SAR 1 N 2 N Tracking Conversion time (log). 12/05/2011-75 ATLCE - D3-2010 DDC

Choices for 12-bit residue ADC Complexity (log) 4095 126 45 28 18 12 Residue 12x1 (Flash) Residue 6x2 <bit/stage>x<num stage> Residue 4x3 Residue 3x4 Residue 2x6 Residue 1x12 1 2 3 4 6 12-- --> Conv. time. 12/05/2011-76 ATLCE - D3-2010 DDC

Telecom applications ADC Direct RF or IF signal conversion Parameters dynamic range (resolution, bit number) conversion rate (1/Tc) linearity (THD) full power bandwidth spurious free dynamic range (SFDR) signal/(noise+distortion) ratio (SINAD) 12/05/2011-77 ATLCE - D3-2010 DDC

Lab experiment Operation and errors of a D/A converter D/A converter with weighted resistors or ladder network, voltage switches, voltage output. Driving with CMOS logic circuits (counter) Measurement of A(D) Evaluation of approximating straight line Gain, offset, nonlinearity errors Conversion in tracking ADC Dynamic range and slew rate verification Text reference sect. 4.L1 12/05/2011-78 ATLCE - D3-2010 DDC

Lesson D3 - final test Which is the effect of strong differential nonlinearity error? Describe the missed-code error. Which parameters can be used to classify ADCs? Draw the block diagram of a residue/subranging converter. Explain the difference between conversion time and latency. How many comparators are required for a 8-bit flash ADC? Which is the conversion delay (as number of comparator decision times) for a 8-bit SAR ADC? Draw the block diagram of a residue converter using 3 stages, with 3 bits each. 12/05/2011-79 ATLCE - D3-2010 DDC