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Lecture 360 – Characterization of ADCs and Sample and Hold Circuits (3/29/10) Page 360-1
LECTURE 360 – CHARACTERIZATION OF ADCS AND SAMPLEAND HOLD CIRCUITSLECTURE ORGANIZATION
Outline• Introduction to ADCs• Static characterization of ADCs• Dynamic characteristics of ADCs• Sample and hold circuits• Design of a sample and hold• SummaryCMOS Analog Circuit Design, 2nd Edition ReferencePages 652-665
Lecture 360 – Characterization of ADCs and Sample and Hold Circuits (3/29/10) Page 360-2
INTRODUCTIONGeneral Block Diagram of an Analog-Digital Converter
DigitalProcessor
Prefilter Sample/Hold Quantizer Encoder
x(t) y(kTN)
Fig.10.5-1
• Prefilter - Avoids the aliasing of high frequency signals back into the baseband of theADC
• Sample-and-hold - Maintains the input analog signal constant during conversion• Quantizer - Finds the subrange that corresponds to the sampled analog input• Encoder - Encoding of the digital bits corresponding to the subrange
Lecture 360 – Characterization of ADCs and Sample and Hold Circuits (3/29/10) Page 360-3
Nyquist Frequency Analog-Digital ConvertersThe sampled nature of the ADC places a practical limit on the bandwidth of the input
signal. If the sampling frequency is fS, and fB is the bandwidth of the input signal, thenfB < 0.5fS
which is simply the Nyquistrelationship which states thatto avoid aliasing, thesampling frequency must begreater than twice thehighest signal frequency.
fB-fB 0 f
fB-fB 0 fSfS-fB fS+fB 2fS2fS-fB 2fS+fBf
-fB 0 fS 2fSf
AntialiasingFilter
fS2
fB-fB 0f
fS2
fS2
fS
fS
Continuous time frequency response of the analog input signal.
Sampled data equivalent frequency response where fB < 0.5fS.
Case where fB > 0.5fS causing aliasing.
Use of an antialiasing filter to avoid aliasing.
Fig. 10.5-
Lecture 360 – Characterization of ADCs and Sample and Hold Circuits (3/29/10) Page 360-4
Classification of Analog-Digital ConvertersAnalog-digital converters can be classified by the relationship of fB and 0.5fS and by theirconversion rate.• Nyquist ADCs - ADCs that have fB as close to 0.5fS as possible.• Oversampling ADCs - ADCs that have fB much less than 0.5fS.
Classification of Analog-to-Digital Converter Architectures
ConversionRate Nyquist ADCs Oversampled ADCs
Slow Integrating (Serial) Very high resolution <14-16 bits
MediumSuccessive
Approximation1-bitPipeline Algorithmic
Moderate resolution <10-12 bits
FastFlash Multiple-bit
Pipeline Folding andinterpolating
Low resolution < 6-8 bits
Lecture 360 – Characterization of ADCs and Sample and Hold Circuits (3/29/10) Page 360-5
Definitions• The dynamic range, signal-to-noise ratio (SNR), and the effective number of bits
(ENOB) of the ADC are the same as for the DAC• Resolution of the ADC is the smallest analog change that distinguishable by an ADC.• Quantization Noise is the ±0.5LSB uncertainty between the infinite resolution
characteristic and the actual characteristic.• Offset Error is the difference between the ideal finite resolution characteristic and
actual finite resolution characteristic• Gain Error is the
difference betweenthe ideal finiteresolution charact-eristic and actualfinite resolutioncharacteristicmeasured at full-scale input. Thisdifference isproportional to theanalog inputvoltage.
000
001
010
011
100
101
110
111
vinVREF
Dig
ital O
utpu
t Cod
e
Offset = 1.5 LSBs
000
001
010
011
100
101
110
111
08
18
28
38
48
58
68
78
88
vinVREF
Dig
ital O
utpu
t Cod
e
Gain Error = 1.5LSBs
(a.) (b.)Figure 10.5-4 - (a.) Example of offset error for a 3-bit ADC. (b.) Example of gainerror for a 3-bit ADC.
IdealCharacteristic
IdealCharacteristic
08
18
28
38
48
58
68
78
88
ActualCharacteristic
Lecture 360 – Characterization of ADCs and Sample and Hold Circuits (3/29/10) Page 360-8
Integral and Differential NonlinearityThe integral and differential nonlinearity of the ADC are referenced to the vertical
(digital) axis of the transfer characteristic.• Integral Nonlinearity (INL) is the maximum difference between the actual finite
resolution characteristic and the ideal finite resolution characteristic measured vertically(% or LSB)
• Differential Nonlinearity (DNL) is a measure of the separation between adjacent levelsmeasured at each vertical step (% or LSB).
DNL = (Dcx - 1) LSBs
where Dcx is the size of the actual vertical step in LSBs.
Note that INL and DNL of an analog-digital converter will be in terms of integers incontrast to the INL and DNL of the digital-analog converter. As the resolution of theADC increases, this restriction becomes insignificant.
Lecture 360 – Characterization of ADCs and Sample and Hold Circuits (3/29/10) Page 360-9
Find the INL and DNL for the 3-bit ADC shown on the previous slide.SolutionWith respect to the digital axis:1.) The largest value of INL for this 3-bit ADC occurs between 3/16 to 5/16 or 7/16 to
9/16 and is 1LSB. 2.) The smallest value of INL occurs
between 11/16 to 12/16 and is-2LSB.
3.) The largest value of DNL occurs at3/16 or 6/8 and is +1LSB.
4.) The smallest value of DNL occursat 9/16 and is -2LSB which iswhere the converter becomesnonmonotonic.
000
001
010
011
100
101
110
111
08
18
28
38
48
58
68
78
88
vinVREF
Dig
ital O
utpu
t Cod
e
DNL =-2 LSB
ActualCharacteristic
IdealCharacteristic
Fig. 10.5-6DL
INL =+1LSB
INL =-2LSB
DNL =+1 LSB
Lecture 360 – Characterization of ADCs and Sample and Hold Circuits (3/29/10) Page 360-12
SAMPLE AND HOLD CIRCUITSRequirements of a Sample and Hold CircuitThe objective of the sample and hold circuit is to sample the unknown analog signal andhold that sample while the ADC decodes the digital equivalent output.The sample and hold circuit must:
1.) Have the accuracy required for the ADC resolution, i.e. accuracy = 100%
2N
2.) The sample and hold circuit must be fast enough to work in a two-phase clock. For anADC with a 100 Megasample/second sample rate, this means that the sample and holdmust perform its function within 5 nanoseconds.3.) Precisely sample the analog signal at the same time for each clock. An advantage ofthe sample and hold circuit is that it removes the precise timing requirements from theADC itself.4.) The power dissipation of the sample and hold circuit must be small. Unfortunately,the above requirements for accuracy and speed will mean that the power must beincreased as the bits are increased and/or the clock period reduced.
Lecture 360 – Characterization of ADCs and Sample and Hold Circuits (3/29/10) Page 360-15
Sample-and-Hold CircuitWaveforms of a sample-and-holdcircuit:
Definitions:• Acquisition time (ta) = time requiredto acquire the analog voltage• Settling time (ts) = time required tosettle to the final held voltage to withinan accuracy tolerance
Tsample = ta + ts Maximum sample rate = fsample(max) = 1
Tsample
Other consideratons:• Aperture time= the time required for the sampling switch to open after the S/Hcommand is initiated• Aperture jitter = variation in the aperture time due to clock variations and noiseTypes of S/H circuits:• No feedback - faster, less accurate• Feedback - slower, more accurate
ta ts
Hold Sample HoldS/H Command
vin*(t)
vin*(t)vin(t)
vin(t)
Time
Am
plitu
de
Fig.10.5-9
Output of S/Hvalid for ADC
conversion
Lecture 360 – Characterization of ADCs and Sample and Hold Circuits (3/29/10) Page 360-16
Settling TimeAssume the op amp has a dominant pole at - a and a second pole at -GB.
The unity-gain response can be approximated as, A(s) GB2
s2 + GB·s + GB2
The resulting step response is, vout(t) = 1 - 43 e-0.5GB·t sin
34 GB·t +
Defining the error as the difference between the final normalized value and vout(t), gives,
Error(t) = = 1 - vout(t) = 43 e-0.5GB·t
In most ADCs, the error is equal to ±0.5LSB. Since the voltage is normalized,1
2N+1 = 43 e-0.5GB·ts e-0.5GB·ts =
43 2N
Solving for the time, ts, required to settle with ±0.5LSB from the above equation gives
ts = 2
GB ln43 2N =
1GB [1.3863N + 1.6740]
Thus as the resolution of the ADC increases, the settling time for any unity-gain bufferamplifiers will increase. For example, if we are using the open-loop, buffered S/H circuitin a 10 bit ADC, the amount of time required for the unity-gain buffer with a GB of 1MHzto settle to within 10 bit accuracy is 2.473μs.
Lecture 360 – Characterization of ADCs and Sample and Hold Circuits (3/29/10) Page 360-18
Switched capacitor S/H circuitwhich autozeroes the op ampinput offset voltage.
A differential version that avoids large changes at the op amp output
New charge (φ2)
Old charge (φ2)
New charge (φ2)
Old charge (φ2)
Attributes:• Accurate• Signal-dependent feedthrough eliminated by a delayed clock• Differential circuit keeps the output of the op amps constant during the 1 phase
avoiding slew rate limits
Lecture 360 – Characterization of ADCs and Sample and Hold Circuits (3/29/10) Page 360-23
If we assume that vin(t) =Vpsin t, then themaximum slope is equal to
Vp.
Therefore, the value of Vis given as
V = dvindt t = Vp t .
The rms value of this noise is given as
V(rms) = dvindt t =
Vp t
2 .
The aperature jitter can lead to a limitation in the desired dynamic range of an ADC. Forexample, if the aperature jitter of the clock is 100ps, and the input signal is a full scalepeak-to-peak sinusoid at 1MHz, the rms value of noise due to this aperature jitter is111μV(rms) if the value of VREF = 1V.
Analog-DigitalConverter
Clock
AnalogInput
DigitalOutput ΔV
t
vin
Aperature Jitter = ΔtFigure10.5-14 - Illustration of aperature jitter in an ADC.
vin(to)
to
Lecture 360 – Characterization of ADCs and Sample and Hold Circuits (3/29/10) Page 360-25
DESIGN OF A SAMPLE AND HOLD AMPLIFIERSpecificationsAccuracy = 10 bitsClock frequency is 10 MHzPower dissipation 1mWSignal level is from 0 to 1VSlew rate 100V/μs with CL = 1pFUse 0.25μm CMOS
Technology Parameters (Cox = 60.6x10-4 F/m2):
Typical Parameter ValueParameter
SymbolParameter Description
N-Channel P-Channel
Units
VT0Threshold Voltage(VBS = 0)
0.5± 0.15 -0.5 ± 0.15 V
K' Transconductance Para-meter (insaturation)
120.0 ± 10% 25.0 ± 10% μA/V2
Bulk thresholdparameter
0.4 0.6 (V)1/2
Channel lengthmodulation parameter
0.32 (L=Lmin)0.06 (L 2Lmin)
0.56 (L=Lmin)0.08 (L 2Lmin)
(V)-1
2| F| Surface potential at strong inversion 0.7 0.8 V
Lecture 360 – Characterization of ADCs and Sample and Hold Circuits (3/29/10) Page 360-26
Op Amp Design – ContinuedThe previous specifications suggest a
self-compensated op amp. The gain andoutput resistance should be easy to achievewith a cascaded output. A folded-cascodeop amp is proposed for the design. Inorder to have the 0-1V signal range, a p-channel, differential input is selected. Thiswill give the input 0-1V range. The outputwill effectively be 0-1V with the unity gainfeedback around the op amp.
Bias Currents:The 100V/μs slew rate requires I3 = 100μA. Setting I4 = I5 = 125μA gives a powerdissipation of 0.875mW with VDD = 2.5V.
061021-01
VNB1
M4 M5
I6
VPB2
I4 I5
VDD = 2.5V
I7M6 M7
VNB2
M8 M9
M10 M11
+
−vIN vOUT
VPB1
I1 I2
M1 M2
M3
I3
CL
Lecture 360 – Characterization of ADCs and Sample and Hold Circuits (3/29/10) Page 360-28
Op Amp Design – ContinuedWe now need to check the output resistance and the gain to make sure the specificationsare satisfied. Let us choose twice minimum channel length to keep the capacitiveparasitics minimized and not have the output resistance too small. Therefore at quiescentconditions,
Op Amp Bias VoltagesWe also need to design the bias voltages VNB1, VNB2, VPB1 and VPB2. This can be doneusing the following circuit:Note, the W/L of M3, M4 and M7 will be 6 so thata current of 10μA gives 100μA in M3 of the opamp. Also, W/L of M1 and M5 will be 16 so acurrent of 10μA gives 125μA in M4 and M5 of theop amp.If M2 is 4 times larger than M1, which gives a W/Lof 64 for M2. Under these conditions,
I2 = I1 = 1
2ß1R2 R = 106
2·120·16·10 = 5.1k
The extra 40 A brings the power dissipation to
0.975mW which is still in specification.
The W/L of M6 and M8 are designed as follows:
VGS8 = VT + 2VON VGS8 - VT = 0.2V = 2·10
120·(W8/L8) W8L8 = 4.167
VSG6 = |VT| + 2VON VSG6 - |VT| = 0.5V = 2·10
25·(W6/L6) W6L6 = 3.20
061021-02
VDD
VPB1
VPB2
VNB2
VNB1
M1 M2
M3 M4
M5
M6M7
M8
R
10µA 10µA10µA
10µA
Lecture 360 – Characterization of ADCs and Sample and Hold Circuits (3/29/10) Page 360-31
Since the signal amplitude is from 0 to 1V, a single NMOS switch should besatisfactory. The resistance of a minimum size NMOS switch is,
RON(worst case) 1
Kn'(W/L)(VGS-VT) = 106
120(1)(1.5-0.5) = 8.33k
For a CH = 1pf, the time constant is 8 ns. This is too close to the 50 ns so let us increasethe switch size to 0.5μm/0.25μm which gives a time constant of 4ns.Therefore, the W/L ratio of the NMOS switch is 0.5μm/0.25μm and the hold capacitor is1pf.Check the error due to channel injection and clock feedthrough-If we assume the clock that rises and falls in 1ns, then a 0.5μm/0.25μm switch works inthe fast transition region. The channel/clock error can be calculated as:
Verror = -W·CGDO +
Cchannel
2CL
VHT -V
3HT
6U·CL -
W·CGDOCL
(VS+2VT -VL)
Lecture 360 – Characterization of ADCs and Sample and Hold Circuits (3/29/10) Page 360-32
Assuming CGDO is 200x10-12 F/m we can calculate VHT as 0.8131V. Thus,Verror =
-100x10-18+0.5(7.57x10-16)
1x10-12 0.8131-0.105x10-3
15x10-3 - 100x10-18
1x10-12 (1+1-0) = -0.586mV
For a 1volt signal with 10 bit accuracy, the error must be less than 1LSB which is0.967mV. The channel/clock error is close to this value and one may have to considerusing a CMOS switch or a dummy switch to reduce the error.
Lecture 360 – Characterization of ADCs and Sample and Hold Circuits (3/29/10) Page 360-33
Design SummaryAt this point, the analog designer understands the weaknesses and strengths of thedesign. The next steps will not be done but are listed below:1.) Simulation to confirm and explore the hand-calculated performance2.) Layout of the op amp, hold capacitor and switch.3.) Verification of the layout4.) Extraction of the parasitics from the layout5.) Resimulation of the design.6.) Check for sensitivity to ESD and latchup.7.) Select package and include package parasitics in simulation.
Lecture 360 – Characterization of ADCs and Sample and Hold Circuits (3/29/10) Page 360-34
SUMMARY• An ADC is by nature a sampled data circuit (cannot continuously convert analog into
digital)• Two basic types of ADCs are:
- Nyquist – analog bandwidth is as close to the Nyquist frequency as possible- Oversampled – analog bandwidth is much smaller than the Nyquist frequency
• The active components in an ADC are the comparator and the sample and hold circuit
• A sample and hold circuit must have at least the accuracy of 100%/2N
• Sample and hold circuits are divided into two types:- Open loop which are fast but not as accurate- Close loop which are slower but more accurate
• An example of designing a sample and hold amplifier was given to illustrate theelectrical design process for CMOS analog circuits