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

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

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]

Residual Offset of Chopping

image-20240903222425730

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

Then, residual offset is

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

Dynamic Element Matching (DEM)

TODO 📅

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]

Qinwen Fan, Evolution of precision amplifiers

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

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