Precision Techniques

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Autozeroing

offset is sampled and then subtracted from the input

Measure the offset somehow and then subtract it from the input signal

low gain comparator

image-20241023224809158

Residual Noise of Auto-zeroing

image-20240826212343905

image-20240826213958740

pnosie Noise Type: timeaverage

image-20240826214306376

Chopping

offset is modulated away from the signal band and then filtered out

Modulate the offset away from DC and then filter it out

Good: Magically reduces offset, 1/f noise, drift

Bad: But creates switching spikes, chopper ripple and other artifacts …

Chopping in the Frequency Domain

Square-wave Modulation

definition of convolution \(y(t) = x(t)*h(t)= \int_{-\infty}^{\infty} x(\tau)h(t-\tau)d\tau\)

for real signal \(H(j\omega)^*=H(-j\omega)\)​

image-20240903222441433

\[ H(j\hat{\omega})*H(j\hat{\omega}) = \int_{-\infty}^{\infty}H(j\omega)H(j(\hat{\omega}-\omega))d\omega \]

sq_mod.drawio

The Fourier Series of squarewave \(x(t)\) with amplitudes \(\pm 1\), period \(T_0\)

\[ C_n = \left\{ \begin{array}{cl} 0 &\space \ n=0 \\ 0 &\space \ n=\text{even} \\ |\frac{2}{n\pi}| &\space n=\pm 1,\pm 5,\pm9, ... \\ -|\frac{2}{n\pi}| &\space n=\pm 3,\pm 7,\pm11, ... \end{array} \right. \]

The Fourier transform of \(s(t)=x(t)x(t)\), and we know \[\begin{align} S(j2n\omega_0) &= \frac{1}{2\pi}\int X(j(2n\omega_0 -\omega))X(j\omega) d\omega\\ &= \frac{1}{2\pi}\int X(j(\omega-2n\omega_0))X(j\omega) d\omega \end{align}\]

Therefore \(n=0\) \[ S(j0) = \frac{1}{2\pi} (2\pi)^2\cdot \frac{4}{\pi ^2}2\sum_{n=0}^{+\infty}\frac{1}{(2n+1)^2} \delta(\omega) = 2\pi \delta(\omega) \]

if \(n=1\)

\[\begin{align} S(j2\omega_0) &= \frac{1}{2\pi} (2\pi)^2\cdot \frac{4}{\pi ^2}\left(1 - 2\sum_{n=0}^{+\infty}\frac{1}{(2n+1)(2n+3)} \right) \\ &= \frac{1}{2\pi} (2\pi)^2\cdot \frac{4}{\pi ^2}\left(1 - 2\sum_{n=0}^{+\infty}\frac{1}{2}\left[\frac{1}{2n+1}- \frac{1}{2n+3}\right] \right) \\ &= 0 \end{align}\]

image-20241013125713945

\(n=2\) \[\begin{align} \sum &= -\frac{2}{3} + 2\left(\frac{1}{1\times 5}+ \frac{1}{3\times 7}+ \frac{1}{5\times 9} + \frac{1}{7\times 11}+...\right) \\ &= -\frac{2}{3} + 2\cdot \frac{1}{4}\left(\frac{1}{1}-\frac{1}{5}+ \frac{1}{3}- \frac{1}{7}+ \frac{1}{5} - \frac{1}{9} +\frac{1}{7}-\frac{1}{11}+...\right) \\ &= -\frac{2}{3} + 2\cdot \frac{1}{4}\frac{4}{3} = 0 \end{align}\]

That is, the input signal remains the same after chopping or squarewave up/down modulation

EXAMPLE 2.7 in R. E. Ziemer and W. H. Tranter, Principles of Communications, 7th ed., Wiley, 2013 [pdf]

Prove that \(\pi^2/8 = 1 + 1/3^2 + 1/5^2 + 1/7^2 + \cdots\) [https://math.stackexchange.com/a/2348996]

Bandwidth & Gain Accuracy

image-20240903225224732

  • lower effective gain: DC level at the output of the amplifiers is a bit less than what it should be

  • chopping artifacts at the even harmonics: frequency of output is \(2f_{ch}\)

REF. [https://raytroop.github.io/2023/01/01/insight/#rc-charge-and-discharge]

chopping_OTA_limitedBW.drawio

Residual Offset of Chopping

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assume input spikes can be expressed as \[ V_\text{spike}(t) = V_o e^{-\frac{t}{\tau}} \]

Then, residual offset is

\[\begin{align} \overline{V_\text{os}} &= \frac{2\int_0^{T_{ch}/2}V_\text{spike}(t)dt}{T_{ch}} \\ &= 2f_{ch}V_o\int_0^{T_{ch}/2} e^{-\frac{t}{\tau}}dt\\ &= 2f_{ch}V_o\tau\int_0^{T_{ch}/2\tau} e^{-\frac{t}{\tau}}d\frac{t}{\tau} \\ &\approx 2f_{ch}V_o\tau \end{align}\]​

Ripple Cancellation after Chopping

On-chip analog filter is not good enough due to limited cutoff frequency

TODO 📅

Dynamic Element Matching (DEM)

TODO 📅

image-20241112214430191

Galton, Ian. (2010). Why dynamic-element-matching DACs work. Circuits and Systems II: Express Briefs, IEEE Transactions on. 57. 69 - 74. 10.1109/TCSII.2010.2042131. [https://sci-hub.se/10.1109/TCSII.2010.2042131]

KHIEM NGUYEN. Analog Devices Inc, "Practical Dynamic Element Matching Techniques for 3-level Unit Elements" [https://picture.iczhiku.com/resource/eetop/shihEDaaoJjFdCVc.pdf]

reference

C. C. Enz and G. C. Temes, "Circuit techniques for reducing the effects of op-amp imperfections: autozeroing, correlated double sampling, and chopper stabilization," in Proceedings of the IEEE, vol. 84, no. 11, pp. 1584-1614, Nov. 1996, doi: 10.1109/5.542410. [http://www2.ing.unipi.it/~a008309/mat_stud/MIXED/archive/2019/Articles/Offset_canc_Enz_Temes_96.pdf]

Kofi Makinwa. Precision Analog Circuit Design: Coping with Variability, [https://youtu.be/nA_DZtRqrTQ?si=6uyOpJhdnYm3iG9d] [https://youtu.be/uwRpP20Lprc?si=SGPta86jRCdECSob]

Chung-chun Chen, Why Design Challenge in Chopping Offset & Flicker Noise? [https://youtu.be/ydjca2KrXgc?si=2raCIB99vXriMPsq]

-, Why Needs A Low Ripple after Chopping Amplifier for A Very Low DC Offset & Flicker Noise? [https://youtu.be/y7TzJtHE7IA?si=kUeP_ESofVxp3IT_]

Qinwen Fan, Evolution of precision amplifiers

Kofi Makinwa, ISSCC 2007 Dynamic-Offset Cancellation Techniques in CMOS [https://picture.iczhiku.com/resource/eetop/sYkywlkpwIQEKcxb.pdf]

Axel Thomsen, Silicon Laboratories ISSCC2012Visuals-T8: "Managing Offset and Flicker Noise" [slides,transcript]