Introduction to the World of Analogue-to-Digital Conversion
Transcript of Introduction to the World of Analogue-to-Digital Conversion
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Introduction to the World of
Analogue-to-Digital Conversion
Shraga Kraus
Analogue and Mixed Signal
Haifa Research Laboratory
ADC
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Contents
• Introduction to A/D Conversion
• Building Blocks
• Basic Architectures
• More Advanced Architectures
• The ΔΣ Architecture
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Contents
• Introduction to A/D Conversion
• Building Blocks
• Basic Architectures
• More Advanced Architectures
• The ΔΣ Architecture
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Introduction to A/D Conversion
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Types of Signals
• Analogue signal:
– Continuous in time
– Continuous in value (within its range of existence)
• Digital signal:
– Discrete in time
– Discrete in value
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The Two Conversions
• Conversion 1: Sampling
– Continuous time discrete time
– Creates aliasing
• Conversion 2: Quantisation
– Continuous value discrete value
– Creates quantisation noise
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Sampling
• Sampling = multiplication by an impulse train
• Y(t) = X(t) · S(t)
• Ts = sampling interval
• fs = 1/Ts = sampling frequency
t
Ts
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Sampling in the Frequency Domain
• In the frequency domain:
• Y(f) = X(f) * S(f)
• “Aliasing” is evident
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Over/Under Sampling
• Nyquistsampling:
• Over-sampling:
• Under-sampling:
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Information at the Output
• Nyquistsampling:
• Over-sampling:
• Under-sampling:
–fs –fs/2 0 fs/2 fs
–fs –fs/2 0 fs/2 fs
–fs –fs/2 0 fs/2 fs
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“Folding” the Frequency Axis
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Finding the Final Frequency
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Example 1: f = 13 MHz
• fs = 20 MHz
• ADC’s output contains information up to 10 MHz
• 13 MHz folds to 7 MHz
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Example 2: f = 23 MHz
• fs = 20 MHz
• ADC’s output contains information up to 10 MHz
• 23 MHz folds to 3 MHz
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Anti-Aliasing Filter
These interferers can be removed digitally
• Never use your ADC at full Nyquist
• A rule of thumb: over sampling ratio (OSR) of 2 is minimal
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Quantisation
• Δ = LSB
• m = number of bits
• Full scale amplitude:
0
Δ
A
76543210
2
2
m
A
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Quantisation Error
• If the input signal is not synchronised with the sampling clock, the error is uniformly distributed between –Δ/2 and +Δ/2
t
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ADC model
• In this case, the ADC is modelled as a linear system with noise
• The noise comes from the quantisation process
• If we refer the noise to the input:
• At every sampling moment a small noise is added to the input signal, bringing it to the centre of the quantisation range
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A Test Case: Continuous-Wave
• The input signal a sinusoidal wave not synchronised with the sampling clock, with a full-scale amplitude
• Quantisation error distribution is:
–Δ/2 +Δ/20
1/ΔFqn
x
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Signal and Noise Power
• Signal power:
• Quantisation noise power:
2
2 2 2 31 1 22
2 2 2
mm
sigP A
3 22 2
2 2
2 2
1 1 1
3 4 12qn qnP F x x dx x dx
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Signal and Noise Power
• SNR:
• Effective number of bits (ENOB):
2 2 32
2
10
2 32
212
10log 6.02 1.76
msig m
qn
dB
PSNR
P
SNR SNR m
1.76
6.02
dBSNRENOB
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Practical ENOB
• In practice, another noise mechanisms exist in the ADC
• Thermal / shot noise
• Nonlinearity (not strictly a noise, but contributes to non-signal power)
• To get the practical resolution of an ADC, the signal to noise-and-distortion ratio (SNDR) should be derived
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Derivation of ENOB
• For a given SNDR of an ADC, the effective resolution is:
𝐸𝑁𝑂𝐵 =𝑆𝑁𝐷𝑅𝑑𝐵 − 1.76
6.02
• The above is valid for a full-scale sinusoidal continuous wave input
• This test setup is feasible using standard lab equipment
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Simulation Example
• Simulation of an ideal 7-bit ADC
• SNR = 43.8 dB ENOB = 7
Quantisation noise is
approximately white
(as expected from a unifor-
mly distributed noise)
Quantisation noise spectral density is
∆2
12
𝑓𝑠2
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Simulation Example - Oversampling
• Out-of-band noise is filtered out digitally
• OSR = 2 SNR x2 (+3dB) ENOB += ½
𝐸𝑁𝑂𝐵 =𝑆𝑁𝐷𝑅𝑑𝐵 − 1.76
6.02
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What is ½ Bit?
• 6½ decimal digits = 106.5
levels
• 7½ bits (binary digits) = 27.5
levels
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Integral Nonlinearity
Vin
outputcode
76543210
Vref0
INL is the horizontal distance between the actual and ideal curves.
Usually expressed in units of Δ.
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Differential Nonlinearity
Vin
outputcode
76543210
Vref0
DNL is the difference between the actual and ideal step widths.
Usually expressed in units of Δ.
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Nonlinearity Information
• INL and DNL provide a lot of information on what’s going on inside the ADC
• Analysis of the data depends on the structure of the specific ADC being tested
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Clock Jitter
• Jitter = the time domain equivalent of phase noise
• Clock jitter = changes in clock period
• Deterministic / random jitter
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Sampling With Clock Jitter
• Clock jitter results in sampling the wrong value
The higherthe slope of the signal,
the larger the error
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Effect of Clock Jitter
• fsig = input signal frequency
• Tj,RMS = random clock jitter RMS (in unit time)
• SNRj = SNR of an ADC as if clock jitter was the only noise source
𝑆𝑁𝑅𝑗 =1
2𝜋 ∙ 𝑓𝑠𝑖𝑔 ∙ 𝑇𝑗,𝑅𝑀𝑆
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Meaning of Clock Jitter
• Clock jitter is painful in sampling of high frequency signals
• No solution was found to date
𝑆𝑁𝑅𝑗 =1
2𝜋 ∙ 𝑓𝑠𝑖𝑔 ∙ 𝑇𝑗,𝑅𝑀𝑆
SNRj depends only on the input signal frequency, NOT
sampling frequency!
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Lab Characterisation - SNR
Pure Sine
Signal Generator
Signal Generator
ADC
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Lab Characterisation - SNDR
Dual Tone
Signal Generator
Signal Generator
ADC
SFDR
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Contents
• Introduction to A/D Conversion
• Building Blocks
• Basic Architectures
• More Advanced Architectures
• The ΔΣ Architecture
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Building Blocks
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Latched Comparator
• Differential pair– When CLK = ‘1’ the input
difference is amplified
• Regenerative load (positive feedback)– When CLK = ‘0’ the
amplified difference if further enhanced to the rails
– Regardless of the inputsOne of many topologies, from Wikipedia
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Comparator’s Nonidealities
• Offset
– Originates from mismatch in the differential pair
• Noise
– Like in every active circuit
– The input-referred noise may result in incorrect decision for small input difference
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Comparator’s Nonidealities
• Regeneration Time
– The smaller the input signal, the longer the regeneration is
• Metastability
– Occurs when the regeneration is too long
– Adding digital buffers at the output can reduce the probability of metastability
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Comparator’s Nonidealities
• Hysteresis
– A result of residual charge somewhere in the load network
– Some circuit techniques alleviate this effect
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Master-Slave Comparator
• Consists of two cascaded latched comparators
• Reduces the probability of metastability
• Has constant input-to-output delay
• But – the delay is long (½ clock cycle)
inp
inn
outp
outn
CK
inp
inn
outp
outn
CK
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More Information
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Op Amp
• Serves in almost every type of ADC
• Based on a differential pair at the input
– except of low voltage technologies, but this makes no difference for our discussion
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Op Amp’s Nonidealities
• Offset
– Originates from mismatch in the differential pair
• Noise
– Like in every active circuit
• Finite Gain and Bandwidth
– Feedback is imperfect, results in gain error
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Op Amp’s Nonidealities
• Inaccurate Gain
– Due to either gain error or mismatch between the feedback devices
• Settling Time
– May set a limit in several architectures
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More Information
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MOS Switch
• Switch resistance (“ON”) depends on Vin
Vctrl = VDD ; Vctrl = 0
VGS = Vctrl – Vin Vin
RSW
Both
PMOS
NMOS
Both
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What’s the Problem With That?
• Nonlinear RSW results in a nonlinear voltage divider
• Signal is distorted
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Parasitic Capacitances
• The gate overlaps with the areas of source and drain
• In addition, the gate capacitance exists
DS
G
B
COL COL
CG
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Charge Injection
• When φ goes down, the charge in the channel is drained to both sides
Vin CL
φ The voltage on
CL changes!
The exact amount of
charge is input dependent
The path of the charge depends on the values of
VOL, CG, and tfall of the clock
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Bootstrapped Switch
• Keeps VGS constant
• Usually incorporates a charge pump
• May also include a sub circuit for alleviating charge injection effect
From P.E. Allen’s lecture notes, 2010
http://www.aicdesign.org/SCNOTES/2010notes/Lect2UP140_%28100325%29.pdf
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More Information
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Sample & Hold / Track & Hold
• Tracks the input signal when clock is low
• Freezes (“holds”) the input value upon clock rising
• Uses a capacitor to store the analogue information (voltage)
φ
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S&H Nonidealities
• Charge injection by the switch
• Leaky capacitor
• Buffer’s nonidealities (including noise)
• Total noise of the capacitor itself is φ 𝐾𝑇
𝐶𝑉2
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More Information
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Contents
• Introduction to A/D Conversion
• Building Blocks
• Basic Architectures
• More Advanced Architectures
• The ΔΣ Architecture
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Basic Architectures
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Flash ADC
• Nº of comparators = 2m – 1
• Thermometer code output
• Converted to binary by a simple logic
• Fastest topology
VREF
Vin
Thermo- meter
to Binary
Dout <2:0>
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Thermometer Code
• Like a bar graph indicator
• The represented decimal number is the number of ‘1’s.
00000000
00000001
00000011
00000111
00001111
00011111
00111111
01111111
11111111
012345678
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Limitations of the Flash ADC
• Many comparators
– area, power
• Resistor matching
– area
• Input capacitance
– requires a buffer
• Comparators’ offset
– Can generate bubbles
VREF
Vin
Thermo- meter
to Binary
Dout <2:0>
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Nonlinearity of the Flash ADC
• Input buffer
• Resistor mismatch
• Comparators’ offset
• Clock/Vin skew or input S&H
VREF
Vin
Thermo- meter
to Binary
Dout <2:0>
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Folding ADC
• Nº of comparators = 2m/2 + 1 – 2
• Reduces area and power significantly– Compared to flash
• In the blue zones, zeros should be counted rather than ones (in )
vout
vin
VREF
VREF
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Numerical Example
• Consider an 8-bit ADC:
• A flash ADC requires 28 – 1 = 255 comparators
• A folding ADC, with 4-4 structure, requires 2 · (24 – 1) = 30 comparators
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Limitations of the Folding ADC
• Same as flash: the MSB flash, , must be as accurate as the LSB (Δ)
– Resistor matching
– Comparators’ offset
• The folding amplifier, , must be as accurate as the LSB (Δ)
– Transistor matching
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Limitations of the Folding ADC
• The folding amplifier introduces a delay and results in skew between the two flash ADCs
– Significant at high frequencies
– Requires introduction of a S&H at the input
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Nonlinearity of the Folding ADC
• Nonlinearity of the flash ADCs
– In particular
• Nonlinearity of the folding amplifier
• Clock/Vin skew between the two flashes or input S&H
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Single Slope ADC
t
VREF
vin
Start!
• Counter reset to 0 and starts counting• Slope initiates
Stop!
• Comparator’s output flips • Counter stops
S&H
vslope
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Limitations of the Single Slope ADC
• The slope must be calibrated
– Its exact slope is unknown
• The comparator makes a decision around Vin, not around a constant voltage
– If the comparator’s offset is constant for every Vin, this only shifts the range of voltages being quantised
– If the comparator’s offset changes with Vin, this results in nonlinearity
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Limitations of the Single Slope ADC
• The S&H has its own nonidealities
• The slope can be nonlinear
• Incomplete capacitor discharge (“memory effect”)
• It’s Slooooooooooow
– 2m clock cycles / conversion
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Single Slope ADC in Digital Cameras
• A line of ADCs samples a line of pixels simultaneously
• To implement the line:– One slope for the entire line
– One counter for the entire line or a counter for each ADC
– A S&H, a comparator and a register for each ADC
– Most CCDs do not require a S&H, though.
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More Information
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Contents
• Introduction to A/D Conversion
• Building Blocks
• Basic Architectures
• More Advanced Architectures
• The ΔΣ Architecture
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More Advanced Architectures
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The Pipeline ADC
• Starts from the MSB
• Then moves on to the next stage – to extract the next bit
• Meanwhile, the first stage treats the next sample
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Principle of Operation
1. Decide about the MSB
2. According to the decision – magnify the appropriate range (2x, shift to centre)
3. Decide about the next bit; go to 2
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Example
C0x2
‘1’
C1
–VREF/2
then
x2
‘0’
C2
‘1’
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Implementation
• Implemented in switched-capacitor technique
• A single-stage DAC + 2x amplifier (MDAC):
Josh Carnes and Un-Ku Moon, “The effect of switch resistance on pipelined ADC MDAC settling time”, Proc. ISCAS 2006
Value of VREF is set according to the decision of
the previous comparator
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How a Multiplying-DAC Works
• During φ1:• During φ2: 𝑉𝑜𝑢𝑡 = 2𝑉𝑖𝑛 − 𝑉𝑅𝐸𝐹
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Offset in the First Comparator
C0x2
‘0’
C1
–VREF/2
then
x2
‘0’
C2
‘1’
Should be ‘0’!
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Limitations of the Pipeline ADC
• Sensitive to comparator’s offset– Solved by redundancy (next slide)
• Sensitive to op amp’s offset– Solved by correlated double sampling (CDS)
• Charge injection from the switches– Solved by bottom plate sampling
• Gain must be exactly 2x– Limited by capacitor mismatch & op amp’s gain
• Op amp’s DC gain must be high– To reduce gain error
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Redundancy – 1½ Bits Per Stage
–VREF/2
then
x2
‘01’
C2B
‘11’
C2A
C0
–VREF/4
then
x2
‘0’
C1B
C1A
MSB = ‘1’ for sure
B1 = ? We’ll decide
later on
Now we have all the information
for making a decision
We’re in the lower partof the 3 upper values D<2:0> = ‘101
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If a Comparator Has Offset
–VREF/2
then
x2
‘00’
C2B
‘10’
C2A
C0
x2
‘1’
C1B
C1A
MSB = ‘1’ for sure
B1 = ‘0’for sure
LSB = ‘1’ for sure
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Limitations of the Pipeline ADC
• Sensitive to comparator’s offset– Solved by redundancy
• Sensitive to op amp’s offset– Solved by correlated double sampling (CDS)
• Charge injection from the switches– Solved by bottom plate sampling
• Gain must be exactly 2x– Limited by capacitor mismatch & op amp’s gain
• Op amp’s DC gain must be high– To reduce gain error
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Nonlinerity of the Pipeline ADC
• Gain is not exactly 2x
– Capacitor mismatch
– Low DC gain of the op amp (gain error)
– Changes the slope of the transfer function
• Long settling time
– Op amp should settle during a clock period
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Notes on the Pipeline ADC
• Nº of comparators = 2m (due to redundancy)
• Nº of MDACs = m – 1
• 1 clock cycle / conversion
• Propagation delay = m clock cycles
• Requires descent op amps
– Not trivial in contemporary CMOS technologies
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More Information
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The SAR ADC
• Based upon binary search (Successive Approximation Register)
• SAR = ‘10000
• DAC > Vin ?
• SAR = ‘C1000
• DAC > Vin ?
• SAR = ‘CC100
• And so on…
From Wikipedia
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Capacitive SAR ADC
• Capacitors are charged to Vin
• Charge is distributed so that 𝑉− = 𝑉𝑅𝐸𝐹 ∙
𝑆𝐴𝑅
2𝑚 − 1− 𝑉𝑖𝑛 = 𝐷𝐴𝐶 − 𝑉𝑖𝑛
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Limitations of the SAR ADC
• Comparator’s offset– Solved by redundancy
• Charge injection from the switches– The larger the capacitors, the smaller the effect
but the slower the ADC is
• Capacitor mismatch
• Low impedance sources of Vin end Vref are required
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Notes on the SAR ADC
• Only one comparator
• m clock cycles / conversion
– Quite slow
• Exponentially scaled capacitors
– A lot of area
• No op amp (!!!)
– Attractive for contemporary CMOS technologies
– Low power
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More Information
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Contents
• Introduction to A/D Conversion
• Building Blocks
• Basic Architectures
• More Advanced Architectures
• The ΔΣ Architecture
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The ΔΣ Architecture
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The Concept of ΔΣ ADC
• Over sampling
– Relatively high OSR
• Noise shaping
– Quantisation noise is attenuated at the frequencies of interest
– It is amplified, on the other hand, at other frequencies
• Feedback structure
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The 1st Order ΔΣ Modulator
• N = loop resolution [bits]
• M = final resolution [bits]“Quantiser”
Loop Filter
For the sake of clarity we will discuss
low pass ΔΣ ADC only
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Linear Model
• Linear Model:
• Loop equation:
Quantisationnoise of the
quantiser
Loop filter of a low pass ΔΣ
𝑌 = 𝑋𝑧−1 + 𝐸 1 − 𝑧−1
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Linear Model
• Signal Transfer Function (STF):
• Noise Transfer Function (NTF):
• Substituting z = ej·2πf in NTF we get:
𝑆𝑇𝐹 = 𝑌
𝑋𝐸=0
= 𝑧−1
𝑁𝑇𝐹 = 𝑌
𝐸𝑋=0
= 1 − 𝑧−1
𝑁𝑇𝐹 𝑓 2 = 2sin 𝜋𝑓 2
Delay
High pass filter
Indeed, a high pass filter
f is normalized:f / fs
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Noise Transfer Function
Quantisationnoise is
attenuated at low frequencies
Quantisationnoise is
amplified at high frequencies
Quantisationnoise of high
frequencies can be filtered out
digitally
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Example: Ideal 1st Order ΔΣ ADC
OSR = 16
The grey noise is filtered out
digitally(decimation filter)
Here, where our signal resides, the quantisation noise
is significantly attenuated
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Example: Discussion
• In a “standard” Nyquist ADC:
– OSR = 16 ENOB += 2
• A 1-bit ADC with OSR = 16:
– ENOB = 3
• A 1-bit 1st order ΔΣ ADC with OSR = 16:
– ENOB = 6.4
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How Does It Work?
• Recall the scheme of a feedback system:
• The closed loop gain is approximately 1/B
– Every pole in B becomes a zero
– Every zero in B becomes a pole
A
B
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How Does It Work?
• In the signal path, the loop filter serves as the amplifier, and there is unity feedback:
• The signal passes as is, with the integrator limiting its bandwidth
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How Does It Work?
• In the error path, the loop filter serves as the feedback, and there is a unity amplifier:
• The closed loop gain is a high pass filter
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The 2nd Order ΔΣ Modulator
• N = loop resolution [bits]
• M = final resolution [bits]
Two feedback loops
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Linear Model
• Linear Model:
• Loop equation:
𝑌 = 𝑋𝑧−1 + 𝐸 1 − 𝑧−1 2
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Linear Model
• Signal Transfer Function (STF):
• Noise Transfer Function (NTF):
• Substituting z = ej·2πf in NTF we get:
𝑆𝑇𝐹 = 𝑌
𝑋𝐸=0
= 𝑧−1
𝑁𝑇𝐹 = 𝑌
𝐸𝑋=0
= 1 − 𝑧−1 2
𝑁𝑇𝐹 𝑓 2 = 2sin 𝜋𝑓 4
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Noise Transfer Function
Quantisationnoise is
attenuated stronger
OSR x 2 + 1.3 bit (1st order)+ 1.7 bit (2nd order)+ 1.9 bit (3rd order)+ 2.1 bit (4th order)
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Higher Loop Orders
• Higher loop orders are possible. However:
• Up to 2nd order the modulator is unconditionally stable– If the linear model shows that the modulator is stable,
it will be stable for every input signal
• From 3rd order and above the modulator is conditionally stable– Even if the linear model shows that the modulator is
stable, it might become unstable for certain input signals
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More Information
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The Mother of All Rules of Thumb
Never do anything
not understanding
what you’re doing
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Thank You!