Interface Electronics

12 Sigma Delta Oversampling ADC

Prof. Dr. Jörg Vollrath

11 Pipeline ADC

Video Lecture: Oversampling Sigma Delta ADC

Länge:

0:0:0 Oversampling ADC

0:4:36 Sigma Delta ADC Architecture

0:12:38 First order passive Sigma Delta Modulator

0:16:51 Internal voltage calculation

0:17:10 0

0:17:10 Example Sigma Delta

0:19:46 Transfer function

0:22:10 f3dB = 15.9 kHz

0:25:46 Output data sequence

0:29:16 dt/R/C = 0.05

0:33:6 Time sequence of voltages

0:36:6 High level simulator (webpage)

0:42:12 LTSPICE simulation sine

0:48:4 General signal and noise transfer function

0:51:14 Noise Shaping

0:52:48 Digital filter and decimator

0:56:16 Example simulation ramp

1:0:46 Example simulation sine

1:5:46 Discussion understanding and simulation

1:9:21 Increasing Signal to noise with oversampling

Review and Overview

Review:
Overview:

Various ADC architectures
Data converter systems:
Nyquist rate, oversampling, undersampling
Objectives sigma delta converter
Oversampling, filter, aliasing, speed and resolution
History of oversampling
Sigma Delta converter architecture
Passive 1st order sigma delta converter: schematic
Passive 1st order sigma delta converter: SPICE
Passive 1st order sigma delta converter: signal example
Passive 1st order sigma delta converter: RC and filter

History

Nyquist ADC

Oversampling

f_CLK = OSR · 2 · f_bw

Clock frequency is much higher than bandwidth.

Pulse count Modulation (PCM)

Predictive Coding

Quantize difference of the signal

Sigma delta converter

Sigma Delta ADC Architectures

1st order passive Sigma Delta Converters
2nd order active Sigma Delta Converters
Higher order active Sigma Delta Converters
Mashed Sigma Delta Converters

Signal to noise ratio
Transfer function and stability
Voltage range preventing saturation

Digital signal processing: filter, decimation

Questions

How to build a sigma delta ADC?

Literature, Theory:
Baker, Razavi, Holberg, Schreier, Wikipedia, SSCC
LTSPICE simulation
Digital filter
Need for high level simulation
Realization, Measurement
Review, Comparison

Overview

Internal voltages: LTSPICE, high level simulation
Transfer function: Stability and noise shaping (SNR)
Signal to noise ratio and architectures
First order, second order, higher order
Digital filters
Digital time domain circuits and frequency representation
Realization
Example realization
Passive
Active: continous time realization (CT)
Cascaded (MASH)
Active: switched capacitor realization (SC)

First order passive Sigma Delta Modulator

Internal voltage levels:

The voltage V(INT n+1) at clock cycle n+1 is:

\( V_{int n+1} = V_{int n} + \frac{\delta t}{C} \left( \frac{ V_{not(Dout)} - V_{int n}}{R} + \frac{V_{in} - V_{int n}}{R} \right) \)
\( V_{int n+1} = V_{int n} + \frac{\delta t}{C \cdot R} \left( V_{not(Dout)} + V_{in} - 2 \cdot V_{int n} \right) \)

\( \delta t \) is the period of the clock.
\( V_{Dout} \) in a real logic circuit is 0V or VDD.
\( \frac{\delta t}{C \cdot R} = \frac{2 \pi f_{sample}}{f_{CLK}} = \frac{2 \pi }{OSR} \)

These equations are used for a high level simulation.
High level simulation: 1st order sigma delta simulation

Since Vint has to stay between 0V and VDD:
\( \frac{\delta t}{C \cdot R} 2 V_{DD} \lt V_{DD} \)
\( C \cdot R \gt \frac{2}{f_{sample} } \)
\( \frac{1}{C \cdot R} \lt \frac{f_{sample} }{2} \)
The bandwidth limit of RC has to be smaller than f_sample/2.
The bandwidth limit of RC has to be greater than the interested bandwith. f_sample/2/OSR
\( 2 \pi f_{bw} \lt \frac{1}{C \cdot R} \lt \frac{f_{sample} }{2} \)

On the left the difference of V(in) and V(D) is integrated and then compared with V(CMP). The comparator operates with the clock frequency and gives for each clock cycle a new value.
A simple passive first order sigma delta converter uses resistors and capacitances.
This circuit is according to Baker, "CMOS integrated cicuits", Figure 6.4

Example: First order passive Sigma Delta Modulator

The circuit with R = 1 kΩ, C= 20 nF is operated with V_DD = 1 V, V_CMP = 0.5 V and f_S = 1 MHz.
What is the maximum allowed input voltage?
What is the cut-off frequency of the RC input low pass?
Calculate vout(t) for 4µs if vin=0.3V. The starting value for vint=VCM and vout=0V. Assume a constant current through R for a sampling cycle.

t [µs]	0	1	2	3	4
vin	0.3V
vint	0.5V
vout	0V

1st order passive Sigma Delta high level simulation

High level simulation: 1st order sigma delta simulation

Internal voltage simulation: Verification of result from previous calculation
Ramp test: Dead zones and noisy transition
Digital filter realisation: Averaging, SINC, droop, bitwidth
FFT: Signal to noise improvement with OSR and filter

General signal and noise transfer function

A first order sigma delta analog digital converter.

\( V_{out}(f) = V_{in}(f) \frac{A(f)}{1 + B(f) A(f)} + E(f) \frac{1}{1 + B(f) A(f)} \)

\( V_{out}(f) = V_{in}(f) \frac{1}{B(f) + \frac{1}{A(f)}} + E(f) \frac{1}{1 + B(f) A(f)} \)

There is a signal transfer function and a noise transfer function.

\( V_{out}(f) = \left( V_{in}(f) - B(f) V_{out}(f) \right) A(f) + E(f) \)
\( V_{out}(f) = V_{in}(f) A(f) - B(f) V_{out}(f) A(f) + E(f) \)
\( V_{out}(f) \left( 1 + B(f) A(f) \right) = V_{in}(f) A(f) + E(f) \)
\( V_{out}(f) = V_{in}(f) \frac{A(f)}{1 + B(f) A(f)} + E(f) \frac{1}{1 + B(f) A(f)} \)
Signal transfer function (STF):
\( STF(f) = \frac{A(f)}{1 + B(f) A(f)} \)
Noise transfer function (NTF):
\( NTF(f) = \frac{1}{1 + B(f) A(f)} \)

First order passive Sigma Delta Modulator signal and noise transfer function

\( \frac{v_{in} - v_{int}}{R} + \frac{-v_{out} - v_{int}}{R} = i_{int} \)
\( v_{in} - v_{int} - v_{out} - v_{int} = i_{int} R \)
\( v_{in} - v_{out} - 2 \cdot v_{int} = i_{int} R \)
\( v_{in} - v_{out} = i_{int} R + 2 \cdot v_{int}\)
\( i_{int} = v_{int} j \omega C \)
\( v_{int} = \frac{v_{in}-v_{out}}{j \omega C R + 2} \)
v_int = A(f) · (v_in - B(f) v_out)
\( B(f) = 1 \) \( A(f) = \frac{1}{j \omega C R + 2} \)
\( V_{out}(f) = \frac{V_{in}}{C R} \frac{1}{j \omega + \frac{3}{R C}} + E(f) \frac{j \omega + \frac{2}{R C}}{j \omega + \frac{3}{R C}} \)

Comparing the equation for the internal voltage v_int with the general signal and noise transfer function gives:
\( V_{int}(f) = A(f) \cdot ( V_{in} - V_{out} \cdot B(f) ) \)
\( B(f) = 1 \)
\( A(f) = \frac{1}{j \omega C R + 2} \)
\( V_{out}(f) = V_{in} \frac{1}{j \omega C R + 3} + E(f) \frac{j \omega C R + 2}{j \omega C R + 3} \)
\( V_{out}(f) = \frac{V_{in}}{C R} \frac{1}{j \omega + \frac{3}{R C}} + E(f) \frac{j \omega + \frac{2}{R C}}{j \omega + \frac{3}{R C}} \)

Noise shaping

Signal pole: ω=3/(RC)
Noise zero: ω=2/(RC)
Noise pole: ω=3/(RC)

Position f_clk
Position f_bw = 3/(2 π R C)

Noise is digitally filtered.
\( \epsilon_q = \frac{LSB}{\sqrt{12}} \)
\( - 20 log \sqrt{12} = -10.8 dB \)

FFT and Noise shaping

Circuit: SigmaDeltaRC.asc

tCLK = 200ns; fCLK = 5 MHZ
16k points makes 3.2768 ms transient simulation plus initialization.
fx = 1/2/pi/R/C=1/2/pi/100k/10p Hz = 160 kHz
OSR = 9.9
dSNR = 30 log(OSR) = 30 dB
ENOB = 30 dB / 6 dB = 6 Bit
Simulation time: 20 min

Figure: SigmaDeltaRC_Sig200k10p32k.jpg

Digital filter and decimator

SINC Filter
Filter order:
SINC, SINC2, SINC3

The order of the filter has to be one more than the sigma delta modulator to realize the full signal to noise benefit.

A decimating Sinc2 filter has the following schematic with 2 integrators and 2 differentiators:

How to physically realize the filter?
Width of the registers?

Calculating Signal to noise ratio

\( V_{noise,RMS}^{2} = 2 \int_0^{BW}{|NTF(f)|^2|V_{Qe}(f)|^2} df \)
Quantization noise spectral density divides the quantization error with sampling frequency:
\( |V_{Qe}(f)|^2 = \frac{V_{LSB}^{2}}{12 f_{sample}} \)
Transfer function:
\( V_{out}(f) = STF(f) v_{in} + NTF(f) V_{Qe}\)
First order passive modulator :
\( V_{out}(f) = \frac{V_{in}}{C R} \frac{1}{j \omega + \frac{3}{R C}} + \frac{E(f)}{R C} \frac{j \omega R C + 2}{j \omega + \frac{3}{R C}} \)
Since signal transfer function and noise transfer function have the part:
\( STF(f) = \frac{1}{j \omega + \frac{3}{RC}} \)
This cancels out for signal to noise ratio.

Calculating Signal to noise ratio (continued)

\( V_{noise,RMS}^{2} = 2 \frac{V_{LSB}^{2}}{12 f_{sample}} \int_0^{BW}{(2 \pi f R C + 2)^2} df \)
\( V_{noise,RMS}^{2} = 2 \frac{V_{LSB}^{2}}{12 f_{sample}} \left( {(2 \pi R C )^2} \frac{BW^3}{3} + 4 \pi R C \frac{BW^2}{2} + 4 BW \right) \)
Sinc-shaped lowpass filter:
\( BW = \frac{f_{sample}}{2 OSR} \)

First part only: \( V_{noise,RMS}^{2} = \frac{V_{LSB}^{2}}{12 } {(2 \pi R C)^2} \frac{f_{sample}^2}{12 OSR^{3}} \)

\( SNR = 20 log\frac{V_{signal,rms}}{V_{noise, rms}} \)
\( SNR = 6.02 N + 1.76 - 20 log \frac{(2 \pi R C) f_{sample}}{\sqrt{12}} + 20 log OSR^{\frac{3}{2}} \)

\( SNR = 6.02 N + 1.76 - 20 log \frac{(2 \pi R C) f_{sample}}{\sqrt{12}} + 30 log OSR \)
With no oversampling signal to noise ratio decreases by:
\( d SNR = - 20 log \frac{(2 \pi R C ) f_{sample}}{\sqrt{12}} \)
The maximum value for f_RC = 1 / (2 π R C) = 0.5 f _sample
. \( d SNR = -20 log(\frac{2}{\sqrt{12}}) = 4.77 dB \) With larger oversampling rates signal to noise ratio improves by 30 log 2 = 9 dB, 9 dB / 6 dB = 1.5 bits for doubling OSR.

Increasing signal to noise with oversampling

Quantization error:
\( SNR = 6.02 B dB + 1.76 dB\)
Averaging:
\( SNR = 6.02 B dB + 1.76 dB + 10 \cdot log( OSR ) \)
Increased Resolution: \( B = \frac{10 \cdot log( OSR )}{6.02} \)
1st order sigma delta
\( SNR = 6.02 B dB + 1.76 dB - 5.17 dB + 30 \cdot log( OSR ) \)
Increased Resolution: \( B = \frac{30 \cdot log( OSR ) - 5.17 }{6.02} \)
2nd order sigma delta
\( SNR = 6.02 B dB + 1.76 dB - 12.9 dB + 50 \cdot log( OSR ) \)
Increased Resolution: \( B = \frac{50 \cdot log( OSR ) - 12.9 }{6.02} \)
General limit
\( SNR_{max} = 10 \cdot log\left( \frac{3(2n+1)}{2 \pi^{2n}} OSR^{2n+1}\right) dB \)
\( SNR_{max} = 10 \cdot log\left( \frac{3(2n+1)}{2 \pi^{2n}}\right) + 10 \cdot (2 \cdot n + 1) \cdot log\left( OSR\right) dB \)
OSR: oversampling rate
n: order of sigma delta modulator

Graph increasing signal to noise with oversampling

Example:
f_bw = 1 kHz
1. Order 1-bit sigma delta
8-bit resolution desired.

Signal to noise, FFT, transfer characteristic and oversampling

Quantization noise is spread over the number of FFT bins N_FFT:
\( \Delta SNR = 10 log\frac{N_{FFT}}{2} \)
Averaging can improve signal to noise ratio
FFT: Half number of points, oversampling with ratio of OSR = 2, reduction of noise
\( \Delta SNR = 10 log OSR \)
Increase of number of bits: OSR = 2 gives Δ SNR = 3 dB is 1/2 bit

Example: 8-bit signal with 4-bit noise amplitude.
Signal: \( V_{rms} = \frac{V_{amplitude}}{\sqrt{2}} \)
Noise: V_rms = V_PP/6 = V_amplitude/3

Averaging (OSR)	Bits	Max value	ENOB
1	8	255	5
4	10	1023	6
16	12	4095	7
256	16	65355	9

Passive First Order Sigma dDelta Design

Given: f_bw = 10 kHz, f_CLK = 1 MHz, V_DD = 1 V

\( OSR = \frac{f_{CLK}}{2 \cdot f_{bw}} = 50 \)
\( N_{Bit} = \frac{30 log(OSR) - 5.2}{6.02} = 8 \)

Capacitance is determined by thermal noise:
\( C (T = 300K) = 12 \cdot k_B \cdot T \frac{2^{2 B}}{V_{FS}^{2}} = 3 fF\)
k_B: Boltzmann constant (1.38 · 10 ^-23 m²kg s^-2K^-1)
T: absolute temperature in Kelvin
B: number of Bits
V_FS = V_DD: Full scale voltage

Resistance is determined by oversampling rate:
\( \frac{\pi}{OSR} = \frac{1}{f_{CLK} R C}\)
\( R \lt \frac{OSR}{\pi C f_{CLK}} = 16 G \Omega \)

Reasonable C and R values for a discrete realization would be.
C = 0.3 nF, R = 160 k Ω
\( \frac{\delta t}{R C} = \frac{1}{f_{CLK} R C} = 0.0208 \)

Comparator needs high gain and low offset for low maximum input voltage swing:

\( d v_{intmax} = V_{FS} \frac{\delta t}{ C \cdot R} = V_{FS} \frac{2 \pi}{OSR} = 0.12 V \)
\( d v_{intmin} = \frac{V_{FS}}{2^{NBit}} \frac{\delta t}{ C \cdot R} = 0.5 mV \)

High level simulation with these values
dt/R/C = 0.0208
gives:
dvint 20 mV (??)
Sinc2, OSR = 64, DNLmax = 7/20 = 0.35
NFFT = 16k
Signal Periods: 3
1 Bit: Signal: -12 dB, Noise -12 dB
SINC2: Signal -12.04 dB, Noise -59.38 dB, SNR = 47.34, ENOB = 7.57
Signal Periods: 5983, SNR 44.43 dB

OSR = 512 = 8 * 64
SINC2: Signal -12.04 dB, Noise -82.19 dB, SNR = 70.15, ENOB = 11.36
Expected change in SNR = 30 log(8) = 27 dB versus real change of 23 dB

dt/R/C = 0.00208
Same result.

Reference: CMOS Analog Circuit Design, Allen, Holberg

Chapter 10.9: Oversampling Converters

[56] Single loop 7th order 118 dB (19 dB)
[54] Single loop 5th order 20 bit:
Thomsen, Bemades, "A digitally Corrrected 20-bit Delta Sigma Modulator", ISSCC, 194-195, Feb 1994
[60-64] 3rd order: 2-1 MASH
Longo, Copeland, "A 13-bit ISDN-Band Oversampled ADC..", CICC, pp.21.2.1-21.2.4, Jan 1988
Williams, Wooley, "A Third order Sigma-Delta..", JSSC, Vol 29. No. 3, pp.193-202,Mar 1994
Yin, Stubbe, Sansen, "Av16-bit 320 kHz CMOS ADC..", JSSC, Vol. 28, No.6, pp.640-647, June 1993
Rabii, Wooley, " A 1.8-V 0.8um CMOS ADC", JSSC, Vol 32., No. 6, pp.783-796, June 1993
Brandt, Wooley, "50MHz 12b 2MHz ADC", JSSC,Vol. 26, No. 6, pp. 1746-1756, Dec 1991
[65-67] 4th order: 2-2 MASH
Tenhunen, "An oversampled ..", ISCAS, pp. 3279-3282, May 1990
Ritoniemi et. al., "A Stereo Audio..", JSSC, Vol. 29, No. 12, pp. 1514-1523, Dec 1994
Fujimori et al., "A 5-V 111dB dynamic range", JSSC, Vol. 32, No. 3, pp.329-336, Mar 1997
[70] 6th order: 2-2-2 MASH
Dedic, " A sixth order", ISSCC Dig. Tech. Papers, pp. 188-189, Feb 1994
[30] MASH Decimator: Candy,Temes, oversampling Delta-Sigma Data Converters. IEEE Press
INF4420 Projects in analog/mixed signal CMOS design

Practical realization of a 1st order Sigma Delta converter

Real 1st order passive sigma delta modulator

Hardware: Digilent NEXYS 3 board with Xilinx FPGA tCLK = 100MHz.

Objectives and Challenges

Generate high resolution sine signal
Analysis of different R, C components, oversampling and timing
Data transfer and processing

Next: Advanced Sigma Delta converters

13 Advanced Sigma Delta Converters

2023 09 29 Interface Electronics 12 Sigma Delta Oversampling ADC Prof. Dr. Joerg Vollrath