## Abstract

The methods for switched-capacitor (SC) noise analysis published up to this date fall in two groups: one group contains methods suitable for analysis by hand that are not easily applicable to all SC circuits. The other group contains methods that are applicable to all SC circuits, but require matrix manipulations with a computer algebra tool. In this paper, we show a universally applicable hand-analysis method. The main reason why SC noise analysis is so difficult is that noise is sampled on many different capacitors, and when being sampled, its spectrum is aliased. The core idea of making analysis by hand possible is to use an intuitive rather than an algebraic method to derive the continuous-time noise spectra in the different phases. Our method combines charge-equation analysis for the discrete-time aspects with signal-flow-graph analysis for the continuous-time aspects of a circuit. We show in tutorial style how to apply it, and demonstrate that it is very useful for getting insight into SC circuits, deriving simplified expressions, and getting a good correspondence with behavioural simulations using SpectreRF.

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## Notes

Source: Private communication with the authors of [14].

## References

Murmann, B. (2012). Thermal noise in track-and-hold circuits: Analysis and simulation techniques.

*IEEE Solid State Circuits Magazine*,*4*(2), 46–54.Gobet, C.-A., & Knob, A. (1983). Noise analysis of switched capacitor networks.

*IEEE Transactions on Circuits and Systems*,*30*(1), 37–43.Gobet, C.-A. (1981). Spectral distribution of a sampled 1st-order lowpass filtered white noise.

*Electronics Letters*,*17*(19), 720–721.Enz, C., & Temes, G. (1996). Circuit techniques for reducing the effects of op-amp imperfections: Autozeroing, correlated double sampling, and chopper stabilization.

*Proceedings of the IEEE*,*84*(11), 1584–1614.Oliaei, O. (2000). Thermal noise analysis of multi-input SC-integrators for delta-sigma modulator design. In

*Proceedings of the ISCAS*(Vol. 4, pp. 425–428). Geneva: IEEE.Schreier, R., Silva, J., Steensgaard, J., & Temes, G. (2005). Design-oriented estimation of thermal noise in switched-capacitor circuits.

*IEEE Transactions on Circuits and Systems I*,*52*(11), 2358–2368.Liou, M., & Kuo, Y.-L. (1979). Exact analysis of switched capacitor circuits with arbitrary inputs.

*IEEE Transactions on Circuits and Systems*,*26*(4), 213–223.Goette, J., & Gobet, C.-A. (1989). Exact noise analysis of SC circuits and an approximate computer implementation.

*IEEE Transactions on Circuits and Systems*,*36*(4), 508–521.Toth, L., Yusim, I., & Suyama, K. (1999). Noise analysis of ideal switched-capacitor networks.

*IEEE Transactions on Circuits and Systems I*,*46*(3), 349–363.Mason, S. (1956). Feedback theory—further properties of signal flow graphs.

*Proceedings of the IRE*,*44*(7), 920–926.Ochoa, A. (1998). A systematic approach to the analysis of general and feedback circuits and systems using signal flow graphs and driving-point impedance.

*IEEE Transactions on Circuits and Systems II*,*45*(2), 187–195.Schmid, H. (2002). Circuit transposition using signal-flow graphs. In

*IEEE international symposium on circuits and systems, ISCAS 2002*(Vol. 2).Ki, W. H. & Temes, G. (1991). Gain- and offset-compensated switched-capacitor filters. In

*Proceedings of the ISCAS*(pp. 1561–1564).Dastgheib, A., & Murmann, B. (2008). Calculation of total integrated noise in analog circuits.

*IEEE Transactions on Circuits and Systems I*,*55*(10), 2988–2993.

## Acknowledgments

We would like to thank the anonymous reviewers for their valuable comments. They really helped us improve the quality of this paper.

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## Appendices

### Appendix 1: SpectreRF configuration

We did all SpectreRF simulations as pss and pnoise simulations. Differing from [1], we set the maximum AC frequency in the pss simulation setup to a sufficiently high value and then used the fullspectrum option in the pnoise simulation. Then we chose that AC frequency so high that increasing it further did not change the result by much anymore.

All simulations were made with behavioural models that we built such that design variables would decide for all switches and amplifiers whether they were noisy or not noisy. This allowed us to use the corner tool of ADEXL to simulate all noise sources independently and simulate the total noise as well. This was, e.g., used in Sect. 7.4 to simulate the circuit with OpAmp noise only.

### Appendix 2: Second-order noise bandwidth

It is well known that

i.e., that the noise bandwidth of the first-order low-pass filter with pole frequency \(f_p\) is \(\pi /2\cdot f_p\).

Interestingly, the noise bandwidths of both the second-order low-pass and band-pass filters with pole frequency \(f_p\) and pole quality factor \(q_p\) is the same, and very simple. For all \(f_p>0\) and \(q_p>0\), and for both \(c=1\) and \(c=f/f_p\),

Surprisingly, the integral can be solved in closed form for any filter order as long as the poles are single [14]. The general solution for the second order is not shown in [14], but is:^{Footnote 1}

For reasons of convenience for the reader, we also show the third-order solution from [14] (the long fourth-order expression can be found there):

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Schmid, H., Eichelberger, L. & Huber, A. A tutorial to switched-capacitor noise analysis by hand.
*Analog Integr Circ Sig Process* **89**, 249–261 (2016). https://doi.org/10.1007/s10470-016-0806-1

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DOI: https://doi.org/10.1007/s10470-016-0806-1