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Accuracy: Deterministic and Random Errors

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Analog-to-Digital Conversion

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Abstract

Technology is the foundation on which every circuit is built. Over the last four decades, technology developments were dominated by the rapid evolution of CMOS technology. This chapter examines a few aspects of that advancement that are relevant to the analog-to-digital designer.

A major worry in design is the variation of parameter values. With the improved control over processing, the batch-to-batch variation is largely under control. The remaining variation between otherwise identical components is generally described by “mis-match” parameters. Next to these random phenomena also systematic errors called “offsets” play an increasingly important role. In conventional ICs, analog circuits with a differential operation (e.g. analog-to-digital converters) were already affected by this random parameter spread. Now statistical phenomena influence the choice for architectures. Understanding and mitigating these effects requires statistical means and models.

The modeling of systematic and random effects that originate from physical, electrical, thermal and lithography effects in devices causing intra-die variations is extensively described.

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Notes

  1. 1.

    The author gratefully acknowledges the many contributions and discussions with his colleagues Hans Tuinhout, Nicole Wils, and Maarten Vertregt in Philips/NXP Research to this chapter.

  2. 2.

    It is always better to rely on simple and oversee-able structures than on the ability to extract precisely the parasitics.

  3. 3.

    Choosing a symmetrical definition removes unnecessary terms from the equations. Obviously any other choice will result in equivalent results, after some more paper work.

  4. 4.

    A rough estimate for a technological gradient can be approximated by dividing the difference between the maximum and minimum of the parameter by half of the wafer diameter.

  5. 5.

    Substrate noise is propagated via the substrate resistance and attenuated by local substrate connections in the analog circuit. A high ratio between both works well.

  6. 6.

    The place to check out lithographical developments is the SPIE conference: www.spie.org. Publications are available on the SPIE digital library website.

  7. 7.

    The processing cost is the determining factor.

  8. 8.

    Compare thermal noise: a flat spectrum in the frequency domain and a Gaussian amplitude distribution.

  9. 9.

    For ease of understanding only, a uniformly distributed dopant is assumed, and more complicated distributions must be numerically evaluated.

  10. 10.

    Here the root of the variance σ is meant. This is the parameter describing the mathematical statistical model. For measurements the term standard deviation is often abbreviated to s.d. and labeled “s.”

  11. 11.

    Measurements generate a standard deviation, while the underlying physics model is described by σ. In the following often σ will be used also when a standard deviation is meant.

  12. 12.

    An often proposed “happy-hour” mismatch model explains threshold fluctuation as a form of petrified 1∕f noise. Threshold mismatch is determined by the number of charged ions and even for the smallest size transistors there are still on average > 100 ions. The same transistor shows none or just one or two electron trapping centers. The numbers do not fit. The trapping centers also follow a Poisson distribution, and the effective v1∕ f,rms follows an area scaling law.

  13. 13.

    Your author had to rebuild in the early 1980s his matching test setup three times before it produced reliable data.

  14. 14.

    The contribution of mobility reduction factor θ and source and drain series resistance are next in line.

  15. 15.

    Accuracy means that the standard deviation of a circuit parameter is within 5–10% of the prediction, see Sect. 6.5.9.

  16. 16.

    In the previous century lithographical emitter deviations were dominant.

  17. 17.

    This effect was used to calibrate the offset in opamps.

  18. 18.

    Random skew is a random but time-invariant effect, while jitter is a random time-variant mechanism.

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  1. Helmond, The Netherlands

    Marcel J. M Pelgrom

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Pelgrom, M.J.M. (2022). Accuracy: Deterministic and Random Errors. In: Analog-to-Digital Conversion. Springer, Cham. https://doi.org/10.1007/978-3-030-90808-9_6

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  • DOI: https://doi.org/10.1007/978-3-030-90808-9_6

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