A tutorial to switched-capacitor noise analysis by hand

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.

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

$$B_1=\int _0^{\infty } \left| \frac{1}{j \frac{f}{f_p} +1} \right| ^2 df = \frac{\pi }{2}f_p = \frac{1}{4} \omega _p\;,$$

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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 $c=f/f_p$,

$$B_2=\int _0^{\infty } \left| \frac{c}{-\frac{f^2}{f_p^2} + j \frac{f}{f_p q_p} +1} \right| ^2 df = \frac{\pi }{2}f_p q_p = \frac{1}{4} \omega _p q_p\;.$$

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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:¹

$$I_2=\int _0^{\infty } \left| \frac{n_1s+n_0}{d_2s^2+d_1s+d_0}\right| ^2 df = \frac{1}{4} \frac{n_1^2d_1d_0+n_0^2d_2d_1}{d_2d_1^2d_0}\;.$$

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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):

$$I_3= \int _0^{\infty } \left| \frac{n_2s^2+n_1s+n_0}{d_3s^3+d_2s^2+d_1s+d_0}\right| ^2 df = \frac{1}{4} \frac{n_2^2d_1d_0+n_1^2d_3d_0+n_0^2d_3d_2-2n_2n_0d_3d_0}{d_3\left( d_2d_1-d_3d_0\right) d_0}\;.$$

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