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Continuous Time Sigma DeltaModulators and VCO ADCs
Pieter Rombouts
Electronics and Information Systems Lab.,Ghent University,
Belgium
Pavia, March 2017
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Outline
1 Sigma Delta Modulation
2 Continuous Time Sigma Delta Modulation
3 FoM Confusion
4 VCO ADC
5 Conclusion
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A/D converter: traditional interpretation
UI
converts analog value into digitalnumber of bits n
I quantisation step q:
q = Vref /2n
error within ±q/2staircase I/O
I static nonlinearityI INL or DNL
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A/D converter: other interpretationUI
UI
Q
converts analog signal into digital signalquantisation eror Q
I (white) noise signalI like other noise contributionsI number of bits not essential
F large enoughLeave margin for other noise sources
F effective bits
quantisation noise variance
σ2Q =
q2
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Core concept 1: Oversamplingspectrum
spectrum
0.1 0.2 0.3 0.4 0.5
signal
white noise
0.1 0.2 0.3 0.4 0.5
frequency/fsample frequency/fsample
Oversampling ratio:
OSR =fS2f0
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Core concept 1: Oversamplingspectrum
spectrum
0.1 0.2 0.3 0.4 0.5
signal
white noise
0.1 0.2 0.3 0.4 0.5
frequency/fsample frequency/fsample
ideal digital filter after quantizerI averaging mechanismI number of bits has increasedI less noiseI filters signal as well
F not Nyquist-rate anymore!
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Core concept 1: Oversamplingspectrum
spectrum
0.1 0.2 0.3 0.4 0.5
signal
white noise
0.1 0.2 0.3 0.4 0.5
frequency/fsample frequency/fsample
quantisation noise variance
σ2Q =
q2
12OSR∼ 1
OSR
3dB/octave improvement
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The Σ∆ control loop
(a) (b)
S quantVin Vin+ +
- -
DH S S
+ D
Q
H
DAC
ideal DACfilter
I discrete timeI continuous time
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The Σ∆ control loop
(a) (b)
S quantVin Vin+ +
- -
DH S S
+ D
Q
H
DAC
D =H
1 + HVin +
1
1 + HQ
for low frequencies H ≈ ∞ −→ D ≈ Vin
nullator
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The Σ∆ control loop
D =H
1 + HVin +
1
1 + HQ︸ ︷︷ ︸
error
input signal is also filtered
for low frequencies NTF = 11+H ≈ 0
for high frequencies NTF = 11+H 6= 0
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Core concept 2: ”Noise” Shaping
NTF(z)
DC
freq.
fsample/2
1
for high frequencies NTF = 11+H 6= 0
spectral shaping
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Core concept 2: “Noise” Shaping
(a) (b)
DC
freq.
fsample/2 DC
freq.
fsample/2
signalshaped noise
noise spectrum has the shape of NTF
combine with oversampling −→ most noise vanishes
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Σ∆ Modulators
(a) (b)
S quantVin Vin+ +
- -
DH S S
+ D
Q
H
DAC
quantizerI very few bitsI accuracy from oversampling + noise shapingI 1 bit
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Σ∆ Modulators
(a) (b)
S quantVin Vin+ +
- -
DH S S
+ D
Q
H
DAC
1-bit quantizerI simpleI inherent linearI noise is not white
F tonesF stability
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Σ∆ Modulators
(a) (b)
S quantVin Vin+ +
- -
DH S S
+ D
Q
H
DAC
multi-bit quantizerI better performanceI DAC needs linearization
F DEMF calibration
I always larger area
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Σ∆ Modulators
(a) (b)
S quantVin Vin+ +
- -
DH S S
+ D
Q
H
DAC
filterI cascade of integratorsI order: design parameter
F trade-off complexity-performanceI special design techniques
F Richard Schreier’s toolbox
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1st order, 1bit Σ∆ ModulatorTypical circuit
+
-C
Vref
Vin
bi
C
+
- biD Q
clk
switched capdevices can be very small
I also CI (thermal) noise ↓ due to oversampling
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High Order Σ∆ ModulatorsCascade of integrators with feedback
DAC
1z -1
b2
1z -1c2
-g1
-a3
b3 b4
-a2
c3
u(n)
v(n)y(n)x3(n)x2(n1z -1
b1
c1x1(n)
-a1
without extra feed insI high swing on internal nodesI ‘poor’ distortion performance
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High Order Σ∆ ModulatorsCascade of integrators with feedforward
DAC
1z -1
b2
1z -1c3
-g1
b3 b4
a3
u(n)
v(n)y(n)x3(n)x2(n1z -1
b1
c2x1(n)
-c1a2
a1
without extra feed insI negligible swing on internal nodesI excellent distortion performance
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Σ∆ ADC
Sigma Deltamodulator
lowpassfilter
Vin anti-aliasingprefilter
f < fcutoff S
fS 2f0 Dout
f = fcutoff 0
analog digital
decimation filter
several filters in chainI simple anti-aliasing filterI no sample-to-sample correspondence
number of bits in Dout high enough
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Σ∆ ADC
Sigma Deltamodulator
lowpassfilter
Vin anti-aliasingprefilter
f < fcutoff S
fS 2f0 Dout
f = fcutoff 0
analog digital
decimation filter
scientific literatureI without filtersI accuracy calculated from ideal filter
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OSR
Low OSR?I keep fs feasableI need many quantizer bitsI need high order filterI minimum 8
High OSR?I small devices
F noise is filtered
I low filter orderI 1-bit quantiser
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Outline
1 Sigma Delta Modulation
2 Continuous Time Sigma Delta Modulation
3 FoM Confusion
4 VCO ADC
5 Conclusion
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CTSDM vs DTSDM
Σ∆ modulatorsI oversampling and noise shapingI high-accuracy
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CTSDM vs DTSDM
Discrete-timeI versatileI “simple” designI easy to “abuse”
standard cell IP core
Continuous-timeI potential for higher speedI potential for lower powerI anti-aliasingI non-trivial design (needs tuning)I performance and stability depend on fclkI common myth: sensitive to clock jitter
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Continuous Time Σ∆ modulator
. . .Vin(s)
fs
Dout(z)
ZOH(s)ZOH(s)
a1sTs
aNsTs
. . .
−Σ Σ
−Σ
−quant
closed feedback loopI cascade of integrators with feedbackI cascade of integrators with feedforward . . .
loop filter = continuous time
sampler inside loop
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Continuous Time Σ∆ modulatorLinearized model
. . .Vin(s)
fs
Dout(z)
ZOH(s)ZOH(s)
a1sTs
aNsTs
. . .
−Σ Σ
−Σ
−Σ
Q
output contains two contributionsI Input signalI Quantisation noise
superposition
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Quantization noise
. . .f s
Dout(z)
ZOH(s)ZOH(s)
a1sTs
aNsTs
. . .
−Σ Σ
−Σ
−Σ
Q
−Σ
ZOH(s)H(s)
f s
Heq(z)
Dout(z)
Q
equivalent discrete time loop filter Heq(z)I fully equivalentI impulse invariant transform of H(s)I CT - DT relationship: z = esTclk
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Quantization noise
−Σ
ZOH(s)H(s)
f s
Heq(z)
Dout(z)
Q
1
Heq + 1· Q = NTF · Q
equivalent to DT Σ∆ modulator
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Quantization noise
−Σ
ZOH(s)H(s)
f s
Heq(z)
Dout(z)
Q
remarksI theory = mature
F e.g. c2d function in matlab
I Heq depends on Dac-pulseI Heq depends on fsI Heq sensitive to analog imperfections
F ‘excess’ loop delayF parasitic (opamp) poles
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Continuous Time Σ∆ modulatorLinearized model
. . .Vin(s)
fs
Dout(z)
ZOH(s)ZOH(s)
a1sTs
aNsTs
. . .
−Σ Σ
−Σ
−Σ
Q
output contains two contributionsI Input signalI Quantisation noise
superposition
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Input signal
. . .Vin(s)
f sDout(z)
ZOH(s)ZOH(s)
a1sTs
aNsTs
. . .
−Σ Σ
−Σ
−
Vin(s) Dout(z)G(s)
f s
−Σ
ZOH(s)H(s)
f s
Heq(z)
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Input signal
Vin(s) Dout(z)G(s)
f s
−Σ
ZOH(s)H(s)
f s
Heq(z)
Vin(s) Dout(z)G(s)
f s
Vin(s) Dout(z)G(s)
f s
AAF(s)
NTF(z)
NTF(z)
equivalent to filter AAF (s) = G (s) ·NTF (z = esTclk)
followed by samplerP. Rombouts (Ghent University) CTSDMs and VCO ADCs Pavia 2017 33 / 80
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Anti-Aliasing in CTSDM
AAF (s) = G (s)NTF (z)
Double filter effect in alias bands (around nfclk)I G (S) lowpass filterI NTF (z) notches at nfclk
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Anti-Aliasing in CTSDM
2−8 2−7 2−6 2−5 2−4 2−3 2−2 2−1 20 21-100
-50
0
50
f/fs
Amplituderespon
se(dB)
NTFeq(esTs)
G(s)
AAF (s)
N = 2a1 = 0.3246a2 = 0.6667
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Feedback vs Feedforward
. . .Vin(s)
fs
Dout(z)
ZOH(s)ZOH(s)
a1sTs
aNsTs
. . .
−Σ Σ
−Σ
−quant
cascade of integrators with feedbackI double anti-aliasing: G (s) and NTF
F less stringent prefiltering requirementsI large internal signal swing
F more demanding opamps
I ADC itself = power hungry but system may be moreefficient
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Feedback vs Feedforward
Vin
-+
+
a1sT
quantansT T
H (s)DAC
a2sT
D+ +
cascade of integrators with feedforwardI single anti-aliasing: NTF but G (s) does not filter
F stringent pre-filtering requirementsI small internal signal swing
F no demanding opamps
I ADC itself = efficient but system may be power hungry
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Noise and power
Vin
-+
+
a1sT
quantansT T
H (s)DAC
a2sT
D+ +
First stage noise dominatesI later stages scaled
F lower powerF still negligible noise
First stage power dominates as wellI increasing order ⇒ moderate impact on power
P. Rombouts (Ghent University) CTSDMs and VCO ADCs Pavia 2017 38 / 80
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Circuit Noise
Ev 2n = 4kTReff · B
noise sees anti-aliasing filterI only in band noiseI no kT/C noise
in theory much better than SC
P. Rombouts (Ghent University) CTSDMs and VCO ADCs Pavia 2017 39 / 80
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‘Excess’ Loop delayCTSDM problems
Vin
-+
+
a1sT
quantansT T
H (s)DAC
a2sT
D+ +
parasitic loop delay
also parasitic poles
P. Rombouts (Ghent University) CTSDMs and VCO ADCs Pavia 2017 40 / 80
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‘Excess’ Loop delayCTSDM problems
Σc1sTs−
Vin(s) Σc2sTs
c3sTs
a3 Σ
fs
Vout(z)
a2
a1d
HDAC(s)
z−1
z−12
−g
e−sτHDAC(s)
parasitic loop delayI make loop delay explicitI add compensation path
also for parasitic poles
P. Rombouts (Ghent University) CTSDMs and VCO ADCs Pavia 2017 41 / 80
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Process variationsCTSDM problems
Σc1sTs−
Vin(s) Σc2sTs
c3sTs
a3 Σ
fs
Vout(z)
a2
a1d
HDAC(s)
z−1
z−12
−g
e−sτHDAC(s)
large errors on RC productsI tuneI robust design
P. Rombouts (Ghent University) CTSDMs and VCO ADCs Pavia 2017 42 / 80
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Slew rateCTSDM problems
opamp not allowed to slewinjection of quantisation noise
I multi-bitI some filtering (e.g. FIR)
P. Rombouts (Ghent University) CTSDMs and VCO ADCs Pavia 2017 43 / 80
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JitterCTSDM problems
Σc1sTs−
Vin(s) Σc2sTs
c3sTs
a3 Σ
fs
Vout(z)
a2
a1d
HDAC(s)
z−1
z−12
−g
e−sτHDAC(s)
jitter in outer feedback DACI directly affects performanceI depends on DAC pulse
P. Rombouts (Ghent University) CTSDMs and VCO ADCs Pavia 2017 44 / 80
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JitterCTSDM problems
clock
ZOH
clock
ZOH
ZOH
white jitterI catastrophicalI solution: multi-bit, FIR etc.
lowpass jitter (= reality)I no big deal
P. Rombouts (Ghent University) CTSDMs and VCO ADCs Pavia 2017 45 / 80
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CTSDM vs DTSDM
Continuous-timeI inherent anti-aliasing
F no noise aliasingcfr kT/C noise in switched cam
F better power-noise trade offI common myth
F sensitive to clock jitter⇒ not as bad as widely assumed
I factsF performance and stability depend on fclkF non-trivial design
P. Rombouts (Ghent University) CTSDMs and VCO ADCs Pavia 2017 46 / 80
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Outline
1 Sigma Delta Modulation
2 Continuous Time Sigma Delta Modulation
3 FoM Confusion
4 VCO ADC
5 Conclusion
P. Rombouts (Ghent University) CTSDMs and VCO ADCs Pavia 2017 47 / 80
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Figure of Merit
Need for FOMI difficult to compare ADC architecturesI different Peak SNDR, Power, Bandwidth, Technology,
areaI which architecture for new design?
P. Rombouts (Ghent University) CTSDMs and VCO ADCs Pavia 2017 48 / 80
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Figure of Merit
Walden’s FOM (1999)
FOMW =P
2ENOB 2 BW
I pJ/conversion code (or pJ/conversion step)I intended to reduce variablesI no justificationI used for many yearsI but meaningless . . .
P. Rombouts (Ghent University) CTSDMs and VCO ADCs Pavia 2017 49 / 80
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Justifying Walden’s FOM?
FOMW =P
2ENOB 2 BW
in good design: P ∼ BWI OK
P ∼ 2N
I ???
P. Rombouts (Ghent University) CTSDMs and VCO ADCs Pavia 2017 50 / 80
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Justifying Walden’s FOM?
P ∼ 2N ???
flash2N comparators
I OK if comparator powerconstant
comparator accuracy ∼ 2−N
I comparator power ∼ 2N
I comparator power ∼ 22N
oops . . .
Vin
+
+
+
+
-
-
-
-
D
D
D
D
Q
Q
Q
Q
clk
clk
clk
clk
Vref
0
1
1
1
P. Rombouts (Ghent University) CTSDMs and VCO ADCs Pavia 2017 51 / 80
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Impedance Scaling laws
I
R
(a)
W
L
C
V
2I
R/2
2W
L
2C
V
(b)
start from best design possibleneed 3dB better SNR
I scale impedances: factor 2
power: factor 4 per bit
P. Rombouts (Ghent University) CTSDMs and VCO ADCs Pavia 2017 52 / 80
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FOM confusion
FOMW =P
2ENOB 2 BW
Walden fixed (scaling laws)
FOMW ,fixed =P
22·ENOB 2 BW
all high accuracy designs were rated bad . . .SAR’s were overrated . . .
P. Rombouts (Ghent University) CTSDMs and VCO ADCs Pavia 2017 53 / 80
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FOM confusion
FOMW ,corrected =P
22·ENOB 2 BW
Walden fixed (scaling laws)I not used
correct FOM: Schreier’s FOM (2005)
FOMS = Peak SNDR + 10log10
(BW
P
)
P. Rombouts (Ghent University) CTSDMs and VCO ADCs Pavia 2017 54 / 80
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Outline
1 Sigma Delta Modulation
2 Continuous Time Sigma Delta Modulation
3 FoM Confusion
4 VCO ADC
5 Conclusion
P. Rombouts (Ghent University) CTSDMs and VCO ADCs Pavia 2017 55 / 80
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VCO-ADC: drivers
++
++
++
Ctrl
N stages Vdd
Ctrl
W
2W
maininverters
auxinverters
4x
4x
1x 1x
vin+
vin-
vout-
vout+
quest for more digital ADC’sI ring oscillatorsI ‘digital’ signalsI no opamps
P. Rombouts (Ghent University) CTSDMs and VCO ADCs Pavia 2017 56 / 80
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VCO-ADC naive
VCO1
Vi n resetcounter
kv fc
D
f sf s f sf s
accuracy ∼ fVCOfs
I e.g. 6-bit for fs = 1GHz, fVCO = 64GHz,
P. Rombouts (Ghent University) CTSDMs and VCO ADCs Pavia 2017 57 / 80
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VCO-ADC equivalent
VCO1
Vi n resetcounter
kv fc
D
f sf s f sf s
VCO1
Vi n counter
kv fcD
f sf s
I1 - z-1
D
P. Rombouts (Ghent University) CTSDMs and VCO ADCs Pavia 2017 58 / 80
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VCO-ADC in phase domain
VCO1
Vi n counter
kv fcD
f sf s
I DVCO1
i n
quant
kv fcD
f sf s
I D
Vi nkv
fc
D
ss
I Di n1/skv
fc
D
sf
s
I
1 - z-1
D+
1 - z-1
phase = integral of frequency
edge occurs when phase = n · 2πphase information is quantized with step = 2π
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VCO-ADC in phase domain
Vi nkv
fc
D
f sf s
I Di n1/skv
fc
D
f sf s
I D+ +
Q
1 - z-1
D(z) ∼(Vin
(1− z−1
)s
)∗+(1− z−1
)︸ ︷︷ ︸NTF
Q
like 1st order CTSDMI anti-aliasingI noise shapingI boost accuracy by oversampling
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multi-phase VCO-ADC
++
++
++
Ctrl
N stages Vdd
Ctrl
W
2W
maininverters
auxinverters
4x
4x
1x 1x
vin+
vin-
vout-
vout+
untill now 1 VCO outputI ring oscillator has N output phases
use all N VCO-phasesI now phase transition at 2π/NI quantization error: N times smallerI higher effective number of bits
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multi-phase VCO-ADC
use all N VCO-phasescounter triggered by N clock inputs?
I use parallelism
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multi-phase VCO-ADC
VCOVi n
resetcounter
D
f sf s f sf s
resetcounter
f sf s f sf s
resetcounter
f sf s f sf s
adder
E.g. N = 64 phases, fs = 1GHz, fVCO = 1GHz,I equivalent fs = 1GHz, fVCO = 64GHz,I 6-bit
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multi-phase VCO-ADC
VCOVi n
resetcounter
D
f sf s f sf s
resetcounter
f sf s f sf s
resetcounter
f sf s f sf s
adder
Reset counter?I special case: 1 bit counterI possible if fVCO ≤ fs
P. Rombouts (Ghent University) CTSDMs and VCO ADCs Pavia 2017 64 / 80
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1-bit counter for VCO-ADCs
clk
D QVCO[i]
DFF
clk
DQ
^
Dout
^
fs
+1
0
+1
0
+1
0
><
(a)
(b)
VCO3
CLK
Dout
TS><TS
t
t
t
><TS
reacts on both edgesI effectively fVCO,eff = 2fVCO
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multi-phase VCO-ADC
w1(t)fff
w2(t)D Q D Q
fs
D Q D Q
fs
D Q D Q
fs
wM(t)
w1(t)
x(t)
+
......
y[n]
M-phases ring oscillator Readout circuit (xM)
2 important properties
P. Rombouts (Ghent University) CTSDMs and VCO ADCs Pavia 2017 66 / 80
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multi-phase VCO-ADC
w1(t)fff
w2(t)D Q D Q
fs
D Q D Q
fs
D Q D Q
fs
wM(t)
w1(t)
x(t)
+
......
y[n]
M-phases ring oscillator Readout circuit (xM)
output = barrel shifted thermometer encodedI inherent DWAI can drive unit element DACI summation = ‘thermometer to binary” coder
P. Rombouts (Ghent University) CTSDMs and VCO ADCs Pavia 2017 67 / 80
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multi-phase VCO-ADC
w1(t)fff
w2(t)D Q D Q
fs
D Q D Q
fs
D Q D Q
fs
wM(t)
w1(t)
x(t)
+
......
y[n]
M-phases ring oscillator Readout circuit (xM)
condition on fVCO :I 0 < fVCO < fs/2
typical sizing:I free running fVCO,0 = fs/4I KV for full scale swingI some tuning
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VCO-ADC challenges
higher order noise shapingI current research
VCO non-linearity
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VCO-ADC with high-order noise shaping
Current work at UGent‘digital’ 3rd order VCO ADC
I prototype in 65 nm CMOSI 12bits@10MHz (with digital calibration)I 11bits@10MHz (without digital calibration)I 3.5 mWI 0.01 mm2
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VCO-ADC challenges
higher order noise shapingVCO non-linearity
I best: 11-bit linearity (UGent)I other solutions
F digital (self)-calibrationF swing reductionF embed in Sigma Delta Loop
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Ghent University linear VCO circuitring oscillator VCO non-linearity
++
++
++
Ctrl
N stagesRing-Osc
Vin Ctrl
R1
R2
Vdd
A. Babaie-Fishani and P. Rombouts,
“Highly linear VCO for use in VCO-ADCs,” Electron. Lett. 2016.
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Ghent University linear VCO circuitring oscillator VCO non-linearity
0 0.2 0.4 0.6 0.8 1100
150
200
250
300
350
400
450
500
(a) Input voltage [volt]
VCO
Fre
quen
cy [M
Hz]
0 0.2 0.4 0.6 0.8 1−3
−2
−1
0
1
2
3
(b) Input voltage [volt]VC
O F
requ
ency
erro
r [M
Hz]
almost 12 bit linearity in pseudo differentialconfiguration
some noise penalty
P. Rombouts (Ghent University) CTSDMs and VCO ADCs Pavia 2017 73 / 80
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digital (self)-calibrationring oscillator VCO non-linearity
Doutnon-linearity
f( )oversamplingnoise-shaping
modulator
digitalnon-linearity
correction
f-1( )Ddec
decimationVi n
look-up tableI nonlinearity is smoothI can be very small (11 points excellent results)
some calibration mechanism
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Input swing reductionring oscillator VCO non-linearity
VCO resetcounter
f skv , fc
+−
Vi n
AD
Cf
DAC
f +Dout(z)
0-1 mash structureI aux ADC and DAC
F uncritical
I sensitive to mismatch
P. Rombouts (Ghent University) CTSDMs and VCO ADCs Pavia 2017 75 / 80
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VCO-ADC in sigma delta loopring oscillator VCO non-linearity
VCOresetcounter Dout(z)
f skv , fc
loopfilter
+
-
DAC
Vi n
embed in Sigma Delta LoopI Original work by Perrott’s groupI input signal of VCO still large
for e.g. 2nd order loop filterI 3rd order noise shapingI 2nd order suppression of VCO nonlinearity
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Phase domain VCO-ADCVCO in feedback loop
VCO2
kv , fc
outVCO1
kv , fc
Vin
+
-
PD+- VCO2
kv , fc
VCO1
kv , fc
+
-
PD+-
register
f s
sampler
Dout(z)+−
DAC
Vi n
VCO performs integrationI pseudo differential
Phase detectorI can be largely digital
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Phase domain VCO-ADCVCO in feedback loop
VCO2
kv , fc
VCO1
kv , fc
+
-
PD+-
register
f s
sampler
Dout(z)+−
DAC
Vi n
VCO performs integrationI pseudo differential
Phase detectorI can be largely digitalI can be multi-phase
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VCO-ADC in sigma delta loopring oscillator VCO non-linearity
VCO2
kv , fc
VCO1
kv , fc
+
-
PD+-
register
f s
sampler
Dout(z)
DAC
loopfilter
+
-
Vi n
for e.g. 2nd order loop filterI 3rd order noise shapingI input signal of VCO smallI 3rd order suppression of VCO nonlinearity
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ConclusionReview of
1 Sigma Delta Modulation
2 Continuous Time Sigma Delta Modulation
3 FoM Confusion
4 VCO ADC
5 Conclusion
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