Hochschule Kempten      
Fakultät Elektrotechnik      
Interface Electronics       Fachgebiet Elektronik, Prof. Vollrath      

Interface Electronics

13 Advanced Sigma Delta Oversampling ADC

Prof. Dr. Jörg Vollrath


12 Sigma Delta ADC




Video Lecture: 13 Oversampling Sigma Delta ADC 9.12.2020


Länge:
0:0:0 Oversampling ADC

0:0:47 First order sigma delta

0:2:1 Increasing signal to noise with oversampling

0:8:47 Averaging

0:12:15 Example:100Mhz Clock, 100kHz bandwidth, 10-bit sigma-delta ADC

0:16:45 OSR=fCLK/2/fbw

0:19:27 Graph Increasing number of bits with oversampling

0:24:41 Oversampling with 1000

0:27:27 Digital droop filter

0:32:4 Problems with sigma delta modulator

0:33:7 Duty cycle and overshoot

0:35:17 DC signal and noise signal in band of interest

0:39:7 Switched capacitor circuit

0:42:22 Average current comparison

0:47:17 Simulation of SC circuit

0:49:37 Benefits and drawbacks

0:51:47 2 non overlapping clocks

0:53:27 Switched capcitor integrator

0:54:35 Second order switched capacitor circuit

0:56:12 Stability and Cascaded sigma delta (2-1-MASH)

1:2:37 Details, Comparator E(z),

1:5:7 Analog, digitized analog and digital signals

1:6:47 Integrator and switched capcitor circuit

1:17:12 z-Elements in digital domain

1:18:59 2-1 MASH sigma delta circuit

1:20:47 Jupyter notebook sigma delta

1:23:57 Delay, latency

Review and Overview

Review:

Droop compensating FIR filter



The graphs are generated by 1st order passive sigma delta JavaScript simulator.
Using a pulse input the FFT shows the transfer function of the system.
Droop of the signal can be seen for higher frequencies (blue curve).
The filter can compensate for part of it (green curve).

An droop compensating FIR Filter [ http://www.cypress.com/file/123171/download ] with a function:
\( out(i) = \frac{-1}{K -2} in(i) + \frac{K}{K -2} in(i-1) + \frac{-1}{K -2} in(i-2) \)
can be activated.
This filter function can be implemented in an FPGA employing shift, add, subtract (using 2s complement) and multiply. Fractions are normalized with the word width of the signal. K values can be taken from the link above.
K values up to Fs/16 up to Fs/8 up to Fs/4
sinc1 25.6573 24.65659 21.04096
sinc1 error +/- dB 0.00017 0.0027 0.045
sinc2 13.79721 13.20875 11.12903
sinc2 error +/- dB 0.0004 0.0064 0.104
sinc3 9.843906 9.393567 7.835116
sinc3 error +/- dB 0.0007 0.011 0.177
FIR filter block diagram:
Experiments:
Show transfer function: Number of periods = 0, SINC1, SINC2, + comp FIR

Sigma Delta ADC basic

Input parameters:
Sample frequency: fclk
Bandwidth: fbw
Number of Bits: B

The ratio \( \frac{f_{clk}}{2 \cdot f_{bw}} = OSR = K \) is called oversampling ratio.
The sigma delta converter can have an order of 1 or higher.
The electric circuit has one RC time constant per order.
The ADC needs a filter and a decimator to generate the final output stream.

Depending on the input parameters, the order (RC) and filter has to be found.
High order sigma delta modulators can give low OSR or high resolution needing more complicated filters.

Sigma Delta ADC Examples

Problems with Sigma Delta ADCs

The graphs can be generated by 1st order passive sigma delta JavaScript simulator.
Tone: negative value -0.651 for period generates a DC level.
Ramp test change 50% duty cycle (55%) or 0% overshoot (5%).

A second order passive Sigma Delta ADC

Switched capacitor circuits


Source: Baker, Mixed signal design, Fig2.35

In switched capacitor circuits resistors can be replaced by capacitances.
The frequency of the non overlapping clock determines the value.
Capacitances can be fabricated with high accuracy and are less temperature dependent than resistors.
The simulation is done with ideal switches.

Comparison to resistor:
What is the average current?
dV = Vin - Vout
CI · dV = Q = Iavg · dt
\( I_{avg} = \frac{C_{I} \cdot dV}{dt} = C_{I} f dV = \frac{1}{R} dV = G dV\)
CI f = 1p · 100MHz = 0.1 mS = G

z-domain:
This is a delay element.
Vout(n) · (CI + CF) = Vout(n-1) CF + Vin(n) CI

(CI + CF) Vout(z) = CF z-1 Vout(z) + CI Vin(z)
\( Vout(z) = \frac{CF}{(CI + CF)} z^{-1} Vout(z) + \frac{CI + CF - CF}{CI+CF} Vin(z) \)

\( a = \frac{CF}{CI + CF}\)

\( Vout(z) = a z^{-1} Vout(z) + (1-a) Vin(z) \)
\( Vout(z) (1 - a z^{-1}) = (1-a) Vin(z) \)
\( a = e^{-\frac{1}{ R C_F f}} = e^{-\frac{C_I}{C_F}} \)
\( H_{RC}(z) = \frac{Y(z)}{X(z)} = \frac{1-a}{1-az^{-1}} \)

Reference: Take the RC Low-Pass Filter to the Z-Domain

Switched capacitor Integrator


Capacitance C1 has charge Q1.
Q1(n) = C1 * Vin
Q2(n) = Q2(n-1) + Q1(n-1)
C2 Vout(n) = C2 Vout(n-1) + C1 Vin(n-1)
C2 Vout(z) = C2 z-1 Vout(z) + C1 z-1 Vin(z)
C2 Vout(z)(1- z-1) = C1 z-1 Vin(z)
\( Vout(z) = \frac{C1}{C2} \frac{z^{-1}}{1-z^{-1}} Vin(z) \)
Vout(z) = H(z) · Vin(z)
\( H(z) = \frac{C1}{C2} \frac{z^{-1}}{1-z^{-1}} = \frac{C1}{C2} \frac{1}{z-1}\)

Second order active switched capacitor Sigma Delta modulator


Source: Baker, Mixed signal design, Fig 7.29
First order SC sigma delta:
\( Vout(z) = E(z) + \frac{1}{z-1} (Vin(z) - Vout(z)) \)
\( Vout(z) \left( 1 + \frac{1}{z-1} \right) = E(z) + \frac{1}{z-1} Vin(z) \)
\( Vout(z) \frac{z}{z-1} = E(z) + \frac{1}{z-1} Vin(z) \)
\( Vout(z) = \frac{z-1}{z} E(z) + \frac{1}{z} Vin(z) \)
\( Vout(z) = (1-z^{-1}) E(z) + z^{-1} Vin(z) \)

Second order SC sigma delta:
\( Vout(z) = E(z) + \frac{1}{z-1} (Vop1(z) - Vout(z)) \)
\( Vout(z) = E(z) + \frac{1}{z-1} \left[ \left( Vin(z)-Vout(z)\right)\frac{1}{z-1} - Vout(z) \right] \)
\( Vout(z) (1 + \frac{1}{z-1} + \frac{1}{(z-1)^{2}}) = E(z) + \frac{1}{(z-1)^{2}} Vin(z) \)
\( Vout(z) (\frac{z}{z-1} + \frac{1}{(z-1)^{2}})= E(z) + \frac{1}{(z-1)^{2}} Vin(z) \)
\( Vout(z) \frac{z(z-1) + 1}{(z-1)^{2}} = E(z) + \frac{1}{(z-1)^{2}} Vin(z) \)
\( Vout(z) = \frac{(z-1)^{2}}{z(z-1) + 1} E(z) + \frac{1}{z(z-1) + 1} Vin(z) \)
\( Vout(z) = \frac{(1-z^{-1})^{2}}{(1-z^{-1}) + z^{-2}} E(z) + \frac{z^{-2}}{(1-z^{-1}) + z^{-2}} Vin(z) \)

Second order SC sigma delta:

\( Vout(z) = E(z) + \frac{1}{z-1} Vop1(z) \)
\( Vout(z) = E(z) + \frac{1}{(z-1)^2} \left( Vin(z)-Vout(z)\right) \)
\( Vout(z) (1 + \frac{1}{(z-1)^{2}}) = E(z) + \frac{1}{(z-1)^{2}} Vin(z) \)
\( Vout(z) \frac{(z-1)^{2}+1}{(z-1)^{2}})= E(z) + \frac{1}{(z-1)^{2}} Vin(z) \)
\( Vout(z) = \frac{(z-1)^{2}}{(z-1)^{2}+1} E(z) + \frac{1}{(z-1)^{2}+1} Vin(z) \)
\( Vout(z) = \frac{(1-z^{-1})^{2}}{(1-z^{-1})^{2} + z^{-2}} E(z) + \frac{z^{-2}}{(1-z^{-1})^{2} + z^{-2}} Vin(z) \)

Cascaded (2-1-MASH) Sigma Delta ADC



Idea


Quantify digitalization error further and correct in digital domain.

The signal and noise transfer function for a 2-1 MASH sigma delta modulator can be calculated:
\( Y_{1}(z) = z^{-2}\cdot X(z) + (1-z^{-1})^2\cdot E_{1}(z) \Rightarrow *z^{-1}\)
\( Y_{2}(z) = z^{-1}\cdot ( E_{1}(z)) + (1-z^{-1}) \cdot E_{2}(z) \Rightarrow *(1-z^{-1})^2 \)

\( Y(z) = z^{-1}\cdot Y_{1}(z) - (1-z^{-1})^2\cdot Y_{2}(z)\)
\( Y(z) = z^{-3}\cdot X(z) + z^{-1} \cdot (1-z^{-1})^2\cdot E_{1}(z) - z^{-1}\cdot (1-z^{-1})^2\cdot E_{1}(z) + (1-z^{-1})^{3} \cdot E_{2}(z)\)
The \(E_{1}(z)\) terms cancel out:
\( Y(z) = z^{-3}\cdot X(z) - (1-z^{-1})^{3} \cdot E_{2}(z)\)

Brandt, Wooley, A 50-MHz Multibit Sigma Delta Modulator for 12b 2-MHz A/D Conversion, JSSC, p.1746-1756, Dec 1991

Digital z-Elements

JavaScript, C:
 Dout = Dout1;
 Dout1 = Din;
JavaScript, C:
  Dout = In - Dout1 - Dout2;
  Dout1 = In - Dout2;
  Dout2 = In;
VHDL:
process (CLK)
  if risingedge(CLK) then
     Dout <= Dout1;
     Dout1 <= Din;
  endif;
end;
VHDL:
Dout <= In - Dout1 - Dout2;
process (CLK)
  if risingedge(CLK) then
      Dout1 <= In - Dout2;
      Dout2 <= In;
  endif;
end;
What bit width is needed?
What happens if bit width is increased?

A 2-1 Mash Sigma Delta ADC Circuit

Source: MSD, Baker Fig 7.58

Next

Next: 14 Figure of merit and summary

ToDo:
"Circuit implementation of the transfer function", Akio Kitagawa, Kanazawa University

Sigma Delta iPhyton notebooks

http://www.python-deltasigma.io/

References

"Circuit implementation of the transfer function", Akio Kitagawa, Kanazawa University
Reference[1]
History no
Schematic1. order passive no
SimulationFull system no
Digital Filterz-Plane no
Digital FilterSchematic no
Digital FilterVHDL no
Practical Circuit no
Measurement Results no
Performance Data B, fC, fCLK, P

5b, 12GS/s, 81 mW ADC, El Chammas, Murrmann, VLSI 2010
Candy,J.C.;Temes,G.C.:OversamplingDeltaSigmaDataConverters.Theory,Design and Simulation.IEEEPress,1991

"Continous Time Modulator", Mitteregger, ISSCC 2006,

"Multi-Mode Modulator", Ouzonouv, ISSCC 2007,

"Boser-Wooley Modulator", Boser & Wooley, JSSC 12/1988,

R. Gray, “Spectral analysis of sigma-delta quantization noise”

Understanding Delta-Sigma Data Converters" by Schreier and Temes (ISBN 0-471-46585-2).
CIFB Architecture, Schreier p.115

Understanding Sigma Delta Data converter, Richard Schreier

Python sigma delta resources


http://www.python-deltasigma.io/

Mapping circuit (time domain) to frequency domain representation, CMOS: Mixed-Signal Circuit Design, Second Edition, Baker, Wiley 2009
Analog Devices: Sigma Delta simulator

A first order sigma delta design example with SNR simulation


Analog Devices AD1877, 16Bit Stereo ADC.

Reference: CMOS: Mixed-Signal Circuit Design

  • Chapter 6: Data Converter Design Basics
    • Figure 6.6/6.7: Passive RC, with RC
    • Decimation Figure 6.10
    • Linearity Figure 6.16
  • Chapter 7: Sigma Delta
    • Tones: Chapter 7.1.4
    • Power supply range 7.1.5
    • Limited gain: stability
    • First order practical 7.1.11, Fig 7.22
    • Second order practical 7.2, Fig 7.40
    • MASH Modulator 7.3.6, Fig 7.58

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