Introduction to Sigma Delta Converters P.V. Ananda Mohan Electronics Corporation of India Limited, Bangalore
How to reduce analog part?• Use Sigma-Delta Conversion• Front-end simple active RC Filter• SC/Gm-C Sigma- Delta converter working at high sampling frequency• Digital Decimation Filter using DSP• Scalable with digital technology• Only few OTAs or opamps, one comparator needed, MOS switches needed
MODERN CODEC-FILTER Active RC SIGMA Decimation Filter -DELTA Digitalinput Converter Filter Digital output Over sampling Digital Clock Filter clocks
What is sigma-delta conversion?• Similar to Delta Modulation but can code dc (i.e.Slowly varying signals)• Generates One bit output sequence• Output word is obtained from this sequence by finding the average using a decimator.• Also called “Pulse density modulation”• Also called “over-sampled A/D conversion”• High resolution up to 19 bits• Uses Oversampling and Noise shaping• Trades off accuracy in amplitude with accuracy in time
Advantages• Analog part small area• Over sampling ratio typically 8 to 256.• Megahertz range up to 16 bits• Band-pass Sigma –delta solutions are also available.
Quantization noise of a linear A/D output input ∆/2-∆/2• Difference between staircase and linear is a triangular waveform.
Linear models for Analysis Difference amplifier V2Ain V1 Comparator + z-1 Integrator + - - I bit DAC Integrator or Accumulator Vref en - Integrator Vi z-1/(1-z-1) Vout • Vo(z)= Vi(z)+en.(1-z-1) • Input low-pass and en high-pass transfer function; Shaping the Quantization noise!!!
Quantization noise for an A/D converter =∆ 2 1 ∆2 / = .∫ 2 2 e n ∆ − / 2 eq ∆ .d e q 12If peak to peak is FSR (Full scale range) , RMSvalue is FSR /(2√2). FSR N N SNR = V =2 =2 2 2 12 6 RMS = ∆ FSR 2 2 2 N 12 2 12 N 6 SNRdB = 20. log 2 2 = 6.02 N + 7.78 − 6.02 = (6.02 N +1.76)dB SNR −1.76 ENOB = 6.02
Quantization noise of a Sigma delta converter• Noise shaping function (1-z-1) L• (1-z-1) L = (1-e-jωT) L = (e-jωT/2)L.(ejωT/2 -e-jωT/2)L• |(1-z-1) L | = | (ejωT/2 -e-jωT/2)L | = 2. Sin(πf/fs)• = (2 πf/fs)L (1−z −1) ∆ df 2 1 2L Noise = f ∫ z= ω ej T 12 s 2 L+ (2π) (π) 1 2L ∆. f f 2 L+1 2L .∆. 2 2 = . = 12 2 L +1 f s 12 2 L +1 2L f s
Candy’s Formula 2 ∆ Dynamic Range DR = 2 2 = 3 2 L +1 . .M 2 L+1 π 2 L+ f ∆ (π ) 1 2L 2 2 2L . . f s 12 2 L +1 DRdB − .76 1 ENOB = 6.02
Advantages• (1-z-1)L has L zeros at dc• Signal and Quantization noise are treated differently.• Output word is obtained from a sequence of coarsely sampled input samples.• Analog part small area• Typical oversampling ratio 8 to 256.
• For a Nyquist rate ADC DR2 = 3.22B-1• For Second Order Sigma delta Converter, 3 5 M=16, L=2 DR = . 4 .165 2 π M16, L=3 3 7 DR = . 6 .167 2 π Ratio is 15.6 dB.For M = 256, Ratio is ?????
Decimation filter• Occupies large area and consumes power.• Linear Phase FIR filter can be used.• Comb filters preferred since the input data is one bit wide only.• Can Reduce sampling rate to four times the Nyquist rate.• Lth order Noise shaping function, L+1th order decimator is required.
Decimation filter• Local average can be computed efficiently by by a decimator.• Frequency response k k ( ) 1− z − D 1 sin(πfDT s ) ( 1− z −1) = D . sin(πf T ) s
Decimation filter• Decimation is reduction of sampling rate• Comb filters are used• Fixed coefficient FIR filters• One,Two or Three stages• Some designs use fixed coefficient IIR filters• Second order Sigma delta converter neds third- order decimator filter.
Decimation filter example output• H(z) = (1-z-64)/(1-z-1) input Input Stream Accumulator clock Lower Clock Latch Output word
DECIMATION FILTERS• Second order Decimator• H(z) = (1-z-64)2/(1-z-1)2• Third-Order Decimator• H(z) = (1-z-64)3/(1-z-1)3• Design is slightly involved-Three parallel processors• Coefficient generation is dynamic for both designs
Single Loop Designs• No non linearity of DAC problems: only two levels one and zero.• Quantization noise power is very high and hence Need large over-sampling ratio• Single loop Sigma-delta modulators, gain progressively increases and overloads the comparator.• Delay also. Input change is felt after five stages.• High coefficient spread (large area)
Single Loop Designs• High coefficient sensitivity• Poor stability• All known digital filter structures cascade, direct form, Leap frog can be used.• Single loop Sigma delta modulators reduce integrator gin to achieve stability.
MASH• Leakage is proportional to 1/Av2• Leakage is proportional to σC2
MASH• Only last stage noise ideally remains.• Noise, distortion performance and Power dissipation dependent largely on the first stage leakage.• Digital Noise cancellation circuits.• Output is a word not a bit as in the case of Single stage 1 bit A/D based design.• Complicates digital filters following the Analog blocks.• Linear single bit Quantizer in the first stage• MASH needs to have low leakage, high opamp gain 90dB low voltage applications not easily realizable.
MASH• Leakage of Quantization noise from each stage is because of the finite gain of the OA• Capacitor mismatch also leads to leakage.• Many versions available called as 1-1-1,1- 2-1, etc indicating the order of the loop in each stage.• Upto fifth order modulators built.
Leslie-Singh Architecture 1-bit DAC NN N+1 - - + 1-bit 2z-1-z-2 ADC Z-1 Z-1 N-bit (1-z-1)2 ADC• This Architecture avoids matching problems of DAC in first stage. Uses Two ADCs in effect.
Why Multi-bit Sigma delta converters?• SNR can be improved by using multi-bit without clocking fast, Candy’s formula• Problem of mismatch of resistors/capacitors occurs• Nonlinearity of DAC is troublesome• Number of bits increases exponentially the complexity (number of capacitors/resistors)• Typically restricted to 4 or 5 bits.• Can be used as single stage Multibit or one stage of MASH
Bandpass Sigma DeltaInput Resonator - • Advantage immune to 1/f noise • No need for matching I and Q signals
Band-Pass Sigma delta Modulator 1/2 Y(z) z-1/ z-1/ Compara (1+z-2) (1+z-2) torX(z) -1 1 z-1 z-1 • H(z)=z-2 and N(z)=(1+z-2)2 Transmission zeroes at fs/4, Obtained by z-1 to –z-2
SC solution• SC preferred because of accurate control of integrator gains.• Fully Differential design increases signal swing by two and dynamic range by 6dB.• Common mode signals such as supply lines, substrate are rejected• Charges injected by switches are cancelled.
• First integrator is important regarding noise, linearity, settling behavior since second stage• Folded cascode Opamps recommended.• Nonidealities of comparator undergo noise shaping and hence not very critical.• Comparator can be simple.• Capacitors chosen based on noise requirements.
Stray-Insensitive Inverting Integrator φ1 C2Vi Vo C1 φ2 φ1 φ2
Stray-Insensitive Non-inverting Integrator φ1 C2 Vi Vo C1 φ2 φ1 φ2
Typical Fully Differential SC integrator VREF- VREF+ 2C Φ1 Y YB Φ2 _ YΦ1d C C + YB Φ1d Φ2 YB Φ1 Y 2C VREF+ VREF-
Timing to avoid charge injection Ф1 Ф2 Ф1d Ф2d
Switch Implementation S D G Inverter• Low ON resistance• Clock Feed-through (reduced by NMOS transistor shunted by PMOS transistor)• Effect is to cause dc offset due to aliasing!
Auto Zero-ed Integrator C2 C3 o e e o o e C4 e o - Vi o C1 + Vo• Haug-Maloberti-Temes• Cancels noise, offset and finite gain effects• Output held over a clock period.
Switch Non-idealities• Fully-differential circuits recommended• Duplicated hardware; more area• Noise of switch due to ON resistance• kT/C noise, large capacitors need to be used for low noise, noise independent of Switch ON resistance Ron• Charging and discharging time dependent on• SC Sigma delta modulators OA of large bandwidth at least five times the sampling frequency and high gain are required.
COMPARATORS• Similar to OPAMPS but need logic level outputs• Input referred offset of MOS Opamps/comparators is quite high.• Offset compensation mandatory.• 10 bit ADC with 1V signal, accuracy of a comparator is 1mV. Thus, residual offset has to be much smaller than 1mV.
A MOS comparator Combining gain stage and latch VB2 M3 M4 Out- In- M1 M2 Out+ In+ M6 M7 M5 M8 Strobe Strobe bar VB1 M9
Typical Bipolar Comparator Preamplifier Latch VoutInput CK CKINV• Latch is a regenerative (positive feedback ) circuit
Flash Architectures for Multibit Sigma delta converters• Quite fast• Number of comparators needed exponentially increases with bit length.• Resistor ladders needed.• Usually 4 to 5 bit Flash A/D used to reduce area.
Flash D/A converter Imperfections• Integral non-linearity• Differential non-linearity• Ac bowing due to input bias current drawn by comparators.• Comparator kickback noise during transit from latching to tracking
CT Sigma Delta Modulators
CT loop ADC filterInput _ DAC
CT Sigma Delta Modulators• Help to increase the clock frequency• Consume less power• OSR needs to be reduced for high bandwidth applications.• No settling behavior problems.• Relaxed sampling networks• More sensitive to clock jitter• ADC jitter not much trouble• But DAC jitter troublesome. Since it is not noise shaped• Non-zero excess loop delay
CT Sigma Delta Modulators• Large RC time constant variation• Mismatch between analog noise shaping and digital noise shaping• CT Filters several alternative technologies abvailable:• Active RC linear, not tunable• Gm-C less power consumption,High frequency, tunable• MOSFET-C non-linearity, advantage of tunability
• First stage is very important• Mixture of Active RC ,Gm-C used.• First stage Active RC for good linearity• Compensation capacitors not needed for Gm- C since integrator capacitor can compensate the OTA
Sigma delta DAC• More tolerant to component mismatch and circuit non-idealities• More digital• Keeping circuit noise low, and meeting linearity are the challenges.
Sigma Delta DAC MSBfN fS fS Digital Interpolation fiterDigital AnalogInput Low- fS VREF pass filter Digital Code Conversion -VREF 0= 100000 =-1 -VREF fS =M.fN Analog 1= 011111=+1 Output DAC
How to combat Nonlinearity of DAC?• Capacitors/Resistors do have mismatch. Randomize the mismatch.• In DWA, same set of DAC elements are used cyclically and repeatedly under the guide of a single pointer.• The element mismatch errors translate to tones at the DAC output when the DAC input has a periodic pattern.• DEM Logic must be optimized for low delay.
DEM Flash output Thermometer code 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 =5 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 =9 C 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 =3 C 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 =12 1 - C 2 + C 15Sixteen capacitors 16 x x x x X x x x x x x x x x x x x x x x x x x x x x x x x
Power Dissipation• Settling performance of the Opamp decides gm.• Power dissipation is dependent on bias current which is decided by Gm.
General Guidelines• Stability (Overload)• Long strings of ones or zeroes are detected and reset is given to integrators to improve stability.• Stabilization techniques needed e.g. clamping of integrator outputs.• Extensive simulation needed e.g Matlab Sigma Delta Tool Box R.Schrier• Rules of Thumb• Maximum of Magnitude of H(z) shall be <1.5 (Lee’s rule)• More relaxed designs available now: Magnitude of H(z) up to 6.• Idle tones in band
General Guidelines• Thumb rule GB of OA > 2.5FS• Switch resistances can be as low as 150 Ohms.• Offset of Comparator <10mV• Hysteresis of comparator <20mV• Single stage modulators are quite tolerant of nonidealities• Instability means that large not necessarily unbounded states gives poor SNR compared to linear models.• Reducing OBG (Out of band gain) improves stability• Capacitors with low voltage coefficients ensure good linearity.
General Guidelines• OPamps with large dc gain in first stage.• Cancellation of even harmonics feasible by fully differential circuits• Top plates of capacitors to virtual grounds of Opamps• Full switches parallel NMOS and PMOS for input whereas only NMOS for those feeding virtual ground.
General Guidelines• Comparator metastability• Effect of clock jitter is independent of the order or structure of the modulator.• Clock A cos(ωot) becomes due to jitter A cos(ωo(t+αSin(ωt)).This adds to the input and sidebands are formed: ωo+ ω, and ωo- ω) of amplitude A α ωo/2• SNR is affected by A2 ωo2/2
Sigma Delta Frequency Synthesizer Frequency Phase Loop Filter VCO reference Detector Multimodulus frequency dividerDataSequence Sigma-Delta Gaussian Precompensation Modulator Filter Filter
Conclusion• More than 400 papers• IEEE press books• Simulation tools• Months of simulation may be needed to weed out problems.• Several solutions• Applications emerging for 802.11, Blue Tooth, CDMA/GSM/3G handsets
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