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Simple Thermal Noise Estimation of Switched Capacitor Circuits Based on OTAs -- Part I: Amplifiers with Capacitive Feedback

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Pith's one-line read This paper claims that thermal noise in OTA-based switched-capacitor circuits can be estimated from three capacitances read off capacitor-only equivalent circuits, avoiding transfer functions and integrals.

desk verdict A genuinely useful design-oriented noise estimation method for OTA-based switched-capacitor circuits, sound under its stated assumptions, but the GmRon << 1 regime needs a quantitative bound and one real-OTA validation. read the letter →

arxiv 1908.08099 v1 pith:DAIFSGLB submitted 2019-08-21 eess.SP

Christian Enz , Antonino Caizzone , Assim Boukhayma , François Krummenacher This is my paper · ORCID
classification eess.SP
keywords thermalnoiseswitched-capacitorcircuitsBodetheoremkTCoperationaltransconductanceamplifiertrackandholdcapacitivefeedbackestimation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that for switched-capacitor circuits built around operational transconductance amplifiers with capacitive feedback, the thermal noise voltage variance between any two nodes can be estimated directly from three capacitances read off capacitor-only equivalent circuits, with no transfer functions and no integrations. If this method is correct, the dominant sampled thermal noise of an SC amplifier or track-and-hold circuit becomes a design-oriented quantity: a few capacitance inspections and a closed-form algebraic expression. The paper demonstrates the recipe on an autozero SC amplifier and an SC track-and-hold circuit and validates the resulting formulas with transient noise simulations.

What carries the argument

The load-bearing object is the extended Bode theorem expressed by Eq. (9), together with the three capacitor-only inspection circuits that define $C_\infty$, $C'_\infty$, and $C_0$. The mechanism is a decomposition of the OTA noise current source into a part at ambient temperature $T$ that joins the switch noises in a passive network, plus an excess-noise part at temperature $(\gamma/h_{fb}-1)T$ evaluated after the switches are shorted. This reduction converts a cyclostationary sampled-noise problem into three capacitance readings and one algebraic summation.

What would settle it

Run a transient-noise simulation of the SC amplifier of Fig. 3 with $C_2=C_L=0.5$ pF, $C_{in}=20$ fF, gain $|A_v|=8$, and a switch resistance raised so that $G_m R_{on}$ is near 1; the $\gamma=0$ switch-only output noise predicted by Eq. (23) will fall increasingly below the simulated value as $R_{on}$ grows, while the full finite-$R_{on}$ calculation tracks the simulation.

Watch

Extended reading notes

Core claim

On its own terms, the paper's discovery is an extension of Bode's theorem from passive RLC networks to active OTA-based switched-capacitor circuits. The variance of the thermal noise voltage between any two nodes $k$ and $l$ is claimed to be $V_{n(kl)}^2 = k_B T [1/C_\infty(kl) + (\gamma/h_{fb}-1)/C'_\infty(kl) - (\gamma/h_{fb})/C_0(kl)]$, where $C_\infty$ is the capacitance seen when all switches and OTAs are removed, $C'_\infty$ is the capacitance seen when closed switches are shorted and OTAs removed, $C_0$ is the capacitance seen when switches are shorted and OTA outputs are grounded, and $h_{fb}$ is the feedback voltage gain from the OTA output to its input. The OTA's excess thermal noise beyond that of a passive conductance is accounted for by assigning it a noise temperature $(\gamma/h_{fb}-1)T$. The paper derives closed-form noise variances for the SC amplifier and the track-and-hold circuit from this identity and checks them against transient noise simulations.

Load-bearing premise

The derivation assumes the switch on-resistance is much smaller than the OTA's effective output resistance, $G_m R_{on} \ll 1$, so the OTA input node behaves as a pure capacitive divider and the switches can be shorted in the excess-noise circuit.

Editorial extensions

If this is right

  • For the autozero SC amplifier, the total output noise at the end of the amplification phase is $V_{nout}^2 = (k_B T/C_2)(\gamma\beta_{ota}+\beta_{sw})$, with both beta factors explicit ratios of $C_1$, $C_2$, $C_{in}$, and $C_L$, matching the classical transfer-function integration.
  • For the SC track-and-hold circuit, the same formula applies with the OTA term coming only from phase 2 and the switch term from both phases; the phase-2 switch contribution is small but is included, refining earlier track-and-hold noise results.
  • Because the method needs only capacitance ratios, it directly exposes which capacitors set the sampled noise floor, which is the quantity a low-power designer must trade against capacitance area and settling time.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper leaves implicit that the same phase-by-phase three-capacitance recipe should extend to multistage switched-capacitor filters and to each step of a successive-approximation ADC front end, as long as charge transfer between phases is handled the way Part I handles the SC amplifier.
  • The paper's stated condition $G_m R_{on} \ll 1$ doubles as a practical error budget: when high-speed designs push the product toward 1, Eq. (9) will under-predict switch noise, and the size of the deviation can be probed by comparing the formula with a nonzero-$R_{on}$ transient simulation.
  • A natural but unstated stress test is finite OTA DC gain: the derivation assumes an infinite-gain ideal OTA, so quantifying the correction for low-gain inverter-based amplifiers would be a testable next step.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. This paper proposes a design-oriented method for estimating the thermal noise voltage variance at any port of an OTA-based switched-capacitor circuit during a given clock phase. The method extends Bode's theorem to active circuits by modelling the ideal OTA as a VCCS, replacing it by a conductance Gm·hfb (where hfb is the capacitive feedback gain), splitting the OTA noise into a part consistent with the ambient temperature and an excess-noise part, and applying Bode's theorem to two passive capacitor-only equivalent circuits. The resulting Eq. (9) expresses the variance in terms of three capacitances, C∞, C'∞, and C0, which can be read by inspection. The paper derives closed-form expressions for a SC autozero amplifier and a SC track-and-hold circuit, and compares them with ELDO transient-noise simulations and with the classical transfer-function integration of Appendix VI-A.

Significance. The proposed formula is parameter-free and falsifiable: Eq. (9) is derived algebraically with no fitted parameters, and Appendix VI-A independently reproduces the variances by classical transfer-function integration, which is a definite strength. If the validity conditions are respected, this gives circuit designers a simple, inspection-based alternative to noise PSD integration for an important class of SC circuits. The main caveat is that the derivation rests on ideal-OTA and GmRon<<1 assumptions, and the paper's own simulations show the approximation degrading at GmRon≈0.27; the method is therefore best viewed as an accurate design estimate within a regime that should be quantified.

major comments (2)
  1. [Section III, Eqs. (4)-(9) and Fig. 10] The central formula Eq. (9) is derived under the condition Gon >> hfbGm (equivalently GmRon << 1), which is used both to replace the switch conductances by short circuits in Fig. 6(d) and, more fundamentally, to justify the input-node relation V = hfb·Vout in Eq. (4). The paper states this condition but gives no quantitative bound on GmRon for a target accuracy. Fig. 10 shows that for γ=0 the switch-only curve deviates visibly from the full calculation once GmRon reaches about 0.27, and the deviation increases with amplifier gain, so a user of Eq. (9) cannot tell from the stated condition whether the formula is accurate to, say, 5%. The authors should provide a design rule or an approximate correction term for finite GmRon, and should present Eq. (9) as an approximation valid in this regime rather than as an exact equality.
  2. [Section IV.A.2, Figs. 10-12] The validation in Figs. 10-12 is performed with the same ideal VCCS model of the OTA and the same switch model (ideal switch plus series resistor Ron) that is used in the derivation, so the agreement demonstrates internal consistency between Eq. (9) and the classical transfer-function integration of Appendix VI-A, but it does not independently confirm the method for real OTAs with finite output conductance and finite DC gain. Since the abstract and conclusion claim applicability to OTA-based SC circuits without these qualifications, the authors should either add a transistor-level verification or explicitly scope the claim to ideal transconductance amplifiers with GmRon << 1.
minor comments (4)
  1. [Abstract and Conclusion] The abstract and conclusion state that the method allows 'precise estimation' without carrying the ideal-OTA and GmRon<<1 qualifications; please add these qualifications to avoid overstating the validated regime.
  2. [Eq. (24b)] There is a duplicated '∼=' in Eq. (24b): the text shows '∼= ∼=αin + ...', which should be a single '∼=' symbol.
  3. [Section III, before Fig. 5] In the sentence 'The later circuits can now be considered as passive', 'later' should be 'latter'.
  4. [Fig. 10 and Section IV.A.2] The caption and text refer to 'Full calculation with Ron > 0' and 'Simulations Ron > 0'; please clarify that the dashed lines are the classical analytical expressions including nonzero GmRon rather than a separate simulator result.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central Eq. (9) is derived from the external Bode theorem via explicit network transformations and is cross-checked against classical noise integration and transient-noise simulations, with no fitted parameter as an input.

full rationale

The central result Eq. (9) is obtained in Section III by replacing the ideal-OTA VCCS with an equivalent conductance Gm·h_fb, splitting the OTA noise source into a thermal part at temperature T and an excess part at (γ/h_fb − 1)T, and applying the standard Bode theorem (Eqs. (1)–(3), from Bode 1945) to the resulting passive capacitor-only subcircuits. Each capacitance C∞, C′∞, and C0 is defined by the usual sZ(s) limits, and the derivation is written out in the paper rather than imported as a conclusion. The results are then independently re-derived by classical transfer-function/noise-bandwidth integration in Appendix VI-A (Tables II and IV), and Eqs. (11) and (16) are stated to be identical to those classical expressions. The only self-citations ([33], [40]) are used for context or as limit checks, not as the justification for Eq. (9), so they are not load-bearing. The Gm·Ron << 1 assumption used to short the switch conductances in Fig. 6(d) is a validity condition, explicitly acknowledged in the discussion of Fig. 10 and in footnote 1 of Section IV.A.2; it limits accuracy but does not make the prediction an input. Transient-noise simulations use the same ideal VCCS model as the derivation, which confirms internal consistency rather than a fitted match. No parameter is fitted to force agreement, and no prediction is defined in terms of the target variance. Score 1 reflects only the negligible self-referentiality of validating an ideal-OTA model with an ideal-OTA simulator; there is no circular step.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The method rests on standard circuit noise assumptions plus the small-switch-resistance approximation. No free parameters are fitted to simulation data; gamma and Gm are circuit model inputs swept in validation, not extracted to make the formula match. No new physical entities are introduced.

assumptions (6)
  • standard math Bode theorem for passive RLC networks: total noise variance equals kT(1/C_infinity - 1/C_0).
    Invoked as the foundation in Section II, Eq. (1); accepted classical result [34].
  • domain assumption Thermal noise sources are white with PSD 4kBT/R for switches and 4kBT gamma Gm for the OTA.
    Used to build the equivalent circuits in Section III and Figs. 4-6.
  • domain assumption OTA is ideal: infinite DC gain, zero offset, infinite input impedance, modeled as a VCCS.
    Stated in Section III; this allows the output voltage to equal the voltage across C2 during charge transfer.
  • domain assumption Switch on-resistances are much smaller than the inverse OTA transconductance (Gm Ron << 1, Gon >> h_fb Gm).
    Used to define h_fb in Eq. (5) as a pure capacitive divider and to short switch resistances in Fig. 6d; the paper shows deviation when Gm Ron is around 0.27.
  • domain assumption Noise during each clock phase is stationary and is frozen at the switching instant.
    Underlies the transfer of sampled charge from C1 to C2 in Eqs. (12)-(13) and the per-phase variance summation.
  • domain assumption Flicker (1/f) noise is negligible or eliminated by autozeroing.
    Assumed in Section I so that thermal noise is the dominant sampled noise source.

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Cite this review

Pith. "Pith review of Simple Thermal Noise Estimation of Switched Capacitor Circuits Based on OTAs -- Part I: Amplifiers with Capacitive Feedback." pith.science (2026). https://pith.science/paper/DAIFSGLB

@misc{pith2026190808099,
  author       = {Pith},
  title        = {Pith review of: Simple Thermal Noise Estimation of Switched Capacitor Circuits Based on OTAs -- Part I: Amplifiers with Capacitive Feedback},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DAIFSGLB}},
  note         = {Machine review of arXiv:1908.08099}
}
read the original abstract

This paper presents a simple method for estimating the thermal noise voltage variance in passive and active switched-capacitor (SC) circuits using operational transconductance amplifiers (OTA). The proposed method is based on the Bode theorem for passive network which is extended to active circuits based on OTAs with capacitive feedback. It allows for a precise estimation of the thermal noise voltage variance by simple inspection of three equivalent circuits avoiding the calculation of any transfer functions nor integrals. In this Part I, the method is applied to SC amplifiers and track&hold circuits and successfully validated by means of transient noise simulations. Part II extends the application of the method to integrators and active SC filters.

Figures

Figures reproduced from arXiv: 1908.08099 by the authors.

Figure 2
Figure 2. Example of a 1st-order low-pass passive filter. 4kBT /R. The thermal noise variance between any nodes k and l of the passive RLC circuit can be calculated using the Bode theorem without the need for computing any integral by simple inspection of two equivalent circuits. The thermal noise voltage variance at any port is simply given by [34]–[36] V 2 n = kBT ·  1 C∞ − 1 C0  , (1) where kB is the Boltzmann constant a… view at source ↗
Figure 3
Figure 3. SC autozero (AZ) amplifier. This Section presents an extension of the Bode theorem to be used for calculating the thermal noise voltage variance seen at any port of an OTA-based SC circuit during a given phase [PITH_FULL_IMAGE:figures/full_fig_p002_3.png] view at source ↗
Figure 5
Figure 5. Equivalent circuit replacing the VCCS of Fig. 4 by a [PITH_FULL_IMAGE:figures/full_fig_p003_5.png] view at source ↗
Figures from the paper (12 more)
Figure 5
Figure 5. Figure 5: The later circuits can now be considered as passive [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]
Figure 6
Figure 6. Figure 6: Simplified schematic of transconductance [PITH_FULL_IMAGE:figures/full_fig_p004_6.png]
Figure 7
Figure 7. Figure 7: Capacitances calculation for the extended Bode theorem [33]. [PITH_FULL_IMAGE:figures/full_fig_p005_7.png]
Figure 8
Figure 8. Figure 8: SC amplifier equivalent circuit schematics for phase [PITH_FULL_IMAGE:figures/full_fig_p006_8.png]
Figure 9
Figure 9. Figure 9: SC amplifier equivalent circuit schematics for phase [PITH_FULL_IMAGE:figures/full_fig_p006_9.png]
Figure 11
Figure 11. Figure 11: Output noise rms voltage versus αL for γ = 0, 1, 2. 600 500 400 300 200 100 0 Vnout [ µ Vrms ] 0 1 2 3 4 5 γ [-] C1=0.5 pF C1=2 pF Extended Bode theorem Simulations T=300 K C2=CL=0.5 pF Cin=20 fF [PITH_FULL_IMAGE:figures/full_fig_p007_11.png]
Figure 12
Figure 12. Figure 12: Output noise voltage versus γ for C1 = 0.5 pF and C1 = 1 pF. close to the simulation results, confirming the origin of the deviation [PITH_FULL_IMAGE:figures/full_fig_p007_12.png]
Figure 13
Figure 13. Figure 13: SC Track & hold circuit. 1) Analysis: The output voltage is read during phase Φ1 from the hold capacitor CL and the sampled output noise must therefore be calculated at the end of phase Φ2. As in the example above, this circuit presents two non-overlapping phases for …
Figure 14
Figure 14. Figure 14: TH equivalent circuit schematics for phase [PITH_FULL_IMAGE:figures/full_fig_p009_14.png]
Figure 15
Figure 15. Figure 15: TH equivalent circuit schematics for phase [PITH_FULL_IMAGE:figures/full_fig_p009_15.png]
Figure 16
Figure 16. Figure 16: Output noise rms voltage versus αL for γ = 0, 1, 2. Φ2, while the second term is due to the noise coming from the switches during phases Φ1 and Φ2. Note that (33) matches the result presented in [41] except for the second term βsw|Φ2 in (34b) which corresponds to the …
Figure 18
Figure 18. Figure 18: Small-signal equivalent circuit of Fig. 3 for the [PITH_FULL_IMAGE:figures/full_fig_p010_18.png]

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    at the Ecole Polytechnique Federale de Lausanne (EPFL), under the supervision of Prof

    He is currently working toward the Ph.D. at the Ecole Polytechnique Federale de Lausanne (EPFL), under the supervision of Prof. Enz and Dr. Boukhayma, on the subject of ultra-low noise and low power sensors for healthcare. Between 2012 and 2013 he worked in STMicroelectronics ...

    2012

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.