Johnson Noise Suppression in AC-Biased Transition-Edge Sensor Bolometers: A Th\'evenin-Equivalent Circuit Analysis

Tijmen de Haan

REVIEW 4 major objections 4 minor 14 references

Johnson Noise Suppression in AC-Biased Transition-Edge Sensor Bolometers: A Th\'evenin-Equivalent Circuit Analysis

T0 review · 4 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read In AC-biased TES bolometers, electrothermal feedback suppresses Johnson noise from the bolometer but not from external series resistance.

desk verdict Useful qualitative insight about parasitic Johnson noise in AC-biased TES readouts, but the central formula contains a clear algebra error and the power-balance derivation uses an unjustified phase-independent treatment of noise. read the letter →

arxiv 2509.01850 v1 pith:UNH44AAY submitted 2025-09-02 astro-ph.IM

Tijmen de Haan This is my paper · ORCID

classification astro-ph.IM

keywords transition-edgesensorJohnsonnoiseelectrothermalfeedbackTheveninequivalentcircuitnoise-equivalentcurrentfrequency-domainmultiplexingparasiticimpedanceACbias

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

Transition-edge sensor bolometers rely on voltage bias and electrothermal feedback to stabilize operation and suppress noise. This paper asks what happens when the bias circuit is not ideal but includes arbitrary series impedance—resistance and reactance from cabling, multiplexers, and other parasitics. Using a Thevenin-equivalent circuit, the author derives a closed-form expression for the total Johnson-noise-equivalent current. The central result is an asymmetry: feedback suppresses Johnson noise generated inside the bolometer's thermal island, but leaves Johnson noise from the external series resistance essentially unsuppressed at high loop gain. For instrument design, this means parasitic resistance sets a noise floor that no amount of feedback can remove.

What carries the argument

The carrying object is the Thevenin-equivalent circuit: any linear readout circuit collapsed to an ideal AC voltage source in series with a complex impedance z_Thev = R_Thev + iX_Thev, plus the bolometer resistance R_b on the thermal island. The associated electrothermal loop gain L encodes how strongly resistance changes feed back through electrical power dissipation. The main result, Eq. (11), combines the two incoherent Johnson contributions into the total noise-equivalent current; its structure shows different prefactors for island versus external noise, which is what produces the suppression asymmetry.

What would settle it

Measure the current-noise spectral density of a TES as a function of bias voltage while a known series resistor is inserted at the bias line. If the noise floor at high loop gain approaches 8kT_parasitic R_parasitic / |Z|^2 rather than decreasing as 1/(1+L), the asymmetry claim is confirmed; if the parasitic term is suppressed along with the bolometer term, the claim fails. A phase-resolved AC circuit simulation would settle the same question by checking Eq. (11) against a random-phase calculation.

Watch

Extended reading notes

Core claim

The paper's central claim is that in an AC-biased transition-edge sensor, electrothermal feedback suppresses Johnson noise from the bolometer's own resistance but leaves Johnson noise from external series resistance essentially unsuppressed. This asymmetry is derived through a Thevenin-equivalent circuit: the entire readout external to the thermal island is collapsed into an ideal AC source plus a complex series impedance, and only the bolometer resistance and its Johnson source sit on the thermal island governed by the power-balance equation. Solving the linearized equations gives Eq. (11), a closed-form total noise-equivalent-current expression valid for arbitrary parasitic impedance, with

Load-bearing premise

The calculation treats each Johnson noise voltage as a phase-independent white noise amplitude when computing dissipated power, using the scalar rule Re(V_J I) = |I| V_J; if the random phase relative to the bias current actually matters, every suppression factor would need to be recomputed.

Editorial extensions

If this is right

For practical TES readout systems with resistive parasitics, increasing electrothermal loop gain indefinitely does not reduce Johnson noise; the unsuppressed parasitic term sets the floor.
Equation (11) provides a complete noise-equivalent-current formula for arbitrary complex Thevenin impedance, so noise budgets can include parasitic resistance and reactance without idealized-bias assumptions.
When reactance is tuned out and parasitic resistance is small, the series impedance has three first-order effects: it changes detector responsivity, reduces the impedance to noise current, and adds its own Johnson noise.
At high loop gain, the bolometer Johnson noise remains suppressed by roughly (1+L), while the parasitic Johnson noise persists at its full amplitude, so minimizing non-superconducting cables, inductor ESR, and coupling to normal metals is essential.
The finite-frequency extension replaces L by L/(1+iωτ), so the same framework describes noise suppression away from the low-frequency limit.

Reading between the lines

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

The island-versus-external split likely generalizes to other off-island noise sources: voltage noise injected anywhere in the Thevenin impedance should escape the (1+L) suppression, which could be tested by injecting a calibrated noise signal at the bias node.
Crosstalk between multiplexed channels acts through mutual Thevenin impedances; the same derivation could be applied per channel to predict how much crosstalk-induced noise survives feedback.
A direct experimental test: add a known series resistor on the bias line and vary its physical temperature while holding loop gain fixed; if the residual noise scales with the resistor's temperature and resistance, the unsuppressed parasitic term is confirmed.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Bottom line: the qualitative point is worth taking seriously—series resistance in AC-biased TES readouts adds Johnson noise that electrothermal feedback won't suppress at high loop gain—but the paper's central result, Eq. (11), is not established. Solving the paper's own Eq. (7) gives a different expression than Eq. (9): the numerator should be |Z|^2, not R_b^2. That is not a minor typo; it changes the claimed exact result for arbitrary impedance. There is also a more fundamental problem with Eq. (5). The power fluctuation from a noise voltage in the AC linearized circuit depends on the phase of the noise phasor relative to the bias current. The paper just states Re(V_J I) = |I| V_J, treating it as a scalar. That is not a standard small-signal treatment; only the in-phase component of the Johnson noise couples to ETF. A correct derivation would split the noise into quadrature components. Without that, the asymmetry between on-island and off-island noise isn't properly derived.

What's good: the Thevenin framing is clean, the distinction between island and parasitic noise is the right one to draw, and the Taylor expansion in Eq. (12) gives practical guidance for system design. The author knows the literature and the context; the references are appropriate. The qualitative conclusion about parasitic resistance setting a noise floor is likely right and useful for experiments like SPT-3G and LiteBIRD.

But between the algebra error and the phase assumption, the exact formulas in Eqs. (9)–(11) are not trustworthy. The paper also overstates that it made 'no assumptions' about the parasitic impedance. If the author fixes the algebra and redoes the power balance with a proper treatment of noise quadratures, the paper could be salvageable. As it stands, it is not ready for publication. It deserves a serious referee, because the question is important and the groundwork is decent, but I would not cite it in its current form.

Referee Report

4 major / 4 minor

Summary. This manuscript develops a Thevenin-equivalent circuit model for Johnson noise in AC-biased TES bolometers read out through a general series impedance z_Thev = R_Thev + iX_Thev. It derives an expression for the total noise-equivalent current (NEI), Eq. (11), and claims that electrothermal feedback suppresses bolometer Johnson noise by (1+L) while leaving parasitic Johnson noise from R_Thev unsuppressed. The framework is clear and the zero-parasitic limit reproduces the familiar V_J,b/[R_b(1+L)] suppression. I checked the derivation algebraically. The main result does not follow from the paper's own equations: (i) the loop gain defined in Eq. (6) is dimensionful and differs from the standard loop gain in the ideal limit; (ii) solving the paper's own power-balance equation (7) gives factors |Z|^2 in deltaR_b that are absent from Eqs. (9)-(10), changing the numerical prediction substantially (e.g., 0.42 vs 0.54 V_J,b for R_b=1, R_Thev=0.2); (iii) Eq. (5) is inconsistent with Eqs. (3)-(4) for finite reactance; and (iv) the phase-independent treatment of Johnson noise is an unproven shortcut. The qualitative asymmetry claim may survive a corrected derivation for real impedances, but the quantitative content of the paper is not valid as written.

Significance. The intended contribution is practically relevant: frequency-multiplexed TES readouts contain parasitic series impedance, and a closed-form NEI would help analyze systems like SPT-3G and LiteBIRD. The explicit, parameter-free, algebraically checkable nature of the derivation is a genuine strength - it makes the verification above possible. The qualitative message (off-island Johnson noise is not suppressed by ETF at high L) is important and appears correct for purely resistive parasitics. However, the checkability cuts both ways: the printed formulas fail their own algebra check at several load-bearing points, so the quantitative results - the point of the paper - cannot be used.

major comments (4)

[§3, Eq. (6)] The loop gain as printed is dimensionally inconsistent: alpha V^2/(G T_b R_b) is dimensionless, but (R_b^2 - |z_Thev|^2)/|Z|^4 has units of Ohm^-2, so L is dimensionful. In the zero-parasitic limit z_Thev=0 this reduces to L = alpha V^2/(G T_b R_b^3), not the standard loop gain alpha V^2/(G T_b R_b). Yet the reduction claimed after Eq. (9) (deltaI_b = V_J,b/(R_b(1+L))) and the suppression factor (1+L)^2 in Eq. (12) require the standard, dimensionless loop gain. Equation (6) therefore introduces a spurious factor R_b^2/|Z|^2 into every subsequent expression and must be corrected before Eqs. (9)-(12) can be interpreted.
[§4, Eqs. (9)-(10) vs Eq. (7)] Solving the bolometer-noise case of Eq. (7) (V_J,Thev=0) gives deltaR_b = L/(1+L) * V_J,b/V * R_b^2(R_b - R_Thev)|Z|^2/(R_b^2 - |z_Thev|^2); the factor |Z|^2 is missing in the printed deltaR_b, and consequently Eq. (9) has 1/Z inside the bracket where the exact solution gives |Z|^2/Z. The parasitic case Eq. (10) has the same defect. A numerical check for R_b = 1 Ohm, R_Thev = 0.2 Ohm, X_Thev=0, L=1: printed Eq. (9) gives |deltaI_b| = 0.54 V_J,b, whereas the solution of Eq. (7) gives 0.42 V_J,b. Because Eqs. (9) and (10) are the building blocks of Eq. (11), the advertised main result is not the NEI of the stated model.
[§3, Eq. (5)] For X_Thev != 0, Eq. (5) does not follow from Eqs. (3)-(4). Expanding deltaP_b = Re(deltaV_island I0* + V_island0 deltaI*) with Eq. (4) and using the paper's own substitution Re(V_J I) = |I| V_J yields a parasitic-noise coefficient 2 R_b V_J,Thev V/|Z|^2, whereas Eq. (5) has (R_b - R_Thev) V_J,Thev V/|Z|^2 + V_J,Thev V/|Z|; these agree only when X_Thev=0. The claim following Eq. (11) that no assumptions are made on the parasitic impedance is therefore unsupported; Eq. (11) is only consistent with Eq. (5) in the purely resistive case.
[§3, before Eq. (5)] The substitution 'Re(V_J I) = |I| V_J' is an ad hoc modelling step, not a standard noise operation. For a noise component at omega +/- Omega, the down-converted power beat with the AC bias is Re(V_J I0* e^{i Omega t}), whose amplitude and sign depend on the random relative phase of V_J and I0; the electrothermal feedback response, and hence the surviving current fluctuation, depend on that phase. Replacing the phasor product by a fixed scalar pre-averages the very quantity the feedback acts on. The result should be checked against a two-sideband linearized calculation (complex thermal-electrical responsivity) with the output PSD averaged over the noise phase; a simple series R-L-C parasitic circuit would suffice as a test. This concern is independent of the algebraic errors above: even with corrected coefficients, the factor conventions (amplitudes vs. two-sided PSDs, V_J^2 = 8kTR) are n

minor comments (4)

[Fig. 2] The caption says 'representative system parameters' but no parameter values are given; the figure is not reproducible and the claimed trends cannot be verified.
[Eq. (8) and surrounding text] The text mixes amplitude notation (V_J, 'V_J,b != 0') with spectral densities (V_J,b^2 = 8kT_b R_b). Readers should be told whether all quantities are PSDs (per Hz) or rms amplitudes, and whether densities are one- or two-sided.
[Throughout] Formatting issues: 'Th´ evenin' appears with broken accents; Eq. (6) is rendered ambiguously (missing parentheses make the denominator hard to parse); the symbols NEP/NEI are never explicitly defined.
[After Eq. (11)] The sentence 'No assumptions have been made regarding the magnitude of the parasitic impedance' is misleading given that Eq. (5) fails for X_Thev != 0 (major comment 3) and Eq. (12) explicitly restricts to X approximately 0.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central NEI expression follows algebraically from the stated circuit model, with the loop gain explicitly defined in Eq. (6).

full rationale

The derivation is self-contained. The main result Eq. (11) is obtained by solving the linearized loop equation (2) together with the power-balance equation (7), both written out in full in the paper. The electrothermal feedback loop gain is not imported as a fitted or prior value: Eq. (6) defines L explicitly, and the citation to de Haan [7] only supplies nomenclature. No parameter is fitted to any subset of data, and no output is a renamed input; the high-loop-gain suppression asymmetry is an algebraic consequence of Eqs. (9)-(10) rather than a tautology. The phase-independent treatment of Johnson noise voltages is an explicit modeling assumption, and whether it is correct is a physical/correctness question, not a circularity, because the result does not reduce to that assumption by construction. Self-citations to [7], [8], and [9] are not load-bearing for the analytic claim.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted to data. The derivation depends on the standard small-signal TES model, the Thevenin reduction, the beta=0 approximation, and an ad hoc treatment of Johnson noise phases. The Johnson-noise-phase treatment is the most fragile and is the candidate weakest assumption.

assumptions (5)

domain assumption Johnson noise voltages can be treated as phase-independent white noise amplitudes with Re(V_J I) = |I| V_J in the power balance
Invoked before Eq (5); this is the paper's own modeling of noise phasors, and it is the load-bearing premise for the suppression factors.
domain assumption The TES current dependence of resistance is zero, beta = dR/dI = 0
Stated in the paragraph below Eq (6); valid for many CMB TES devices but not all.
domain assumption Power balance in quasi-equilibrium: delta P_b = G delta T = G T_b delta R_b / (alpha R_b)
Equation (3); assumes a single thermal body with constant dynamic conductance G in the low-frequency limit.
standard math Small-signal linearization: first-order perturbation in noise voltages and resistance change, higher-order terms neglected
Section 'Linearized Analysis'.
standard math Thevenin's theorem applies so the readout reduces to a single voltage source with series impedance
Section 'Thevenin-Equivalent Circuit Representation'.

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Cite this review

Pith. "Pith review of Johnson Noise Suppression in AC-Biased Transition-Edge Sensor Bolometers: A Th\'evenin-Equivalent Circuit Analysis." pith.science (2026). https://pith.science/paper/UNH44AAY

@misc{pith2026250901850,
  author       = {Pith},
  title        = {Pith review of: Johnson Noise Suppression in AC-Biased Transition-Edge Sensor Bolometers: A Th\'evenin-Equivalent Circuit Analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UNH44AAY}},
  note         = {Machine review of arXiv:2509.01850}
}

read the original abstract

Transition-edge sensor (TES) bolometers operate under strong electrothermal feedback, wherein power deposited on the bolometer is compensated by a corresponding change in electrical power dissipation. We present a comprehensive analysis of Johnson noise suppression that extends previous theoretical frameworks to the general case of AC-biased linear circuits with arbitrary parasitic impedances. Using a Th\'evenin-equivalent circuit formulation -- consisting of an ideal voltage source and complex series impedance external to the bolometer thermal island -- we derive analytical expressions for the noise-equivalent current in the presence of both bolometer and parasitic Johnson noise sources. Our analysis demonstrates that while the electrothermal feedback loop effectively suppresses Johnson noise originating from the bolometer resistance, it provides no suppression of Johnson noise from external series resistance. In the limit of high loop gain, the bolometer Johnson noise contribution vanishes while the parasitic contribution remains unsuppressed.

Figures

Figures reproduced from arXiv: 2509.01850 by the authors.

**Figure 2.** FIG. 2. Johnson noise suppression as a function of electrothermal feedback loop gain [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

J. C. Mather, Bolometer noise: nonequilibrium theory, Applied Optics 21, 1125 (1982), publisher: Optica Publishing Group

work page 1982

[2] [2]

S. F. Lee, J. M. Gildemeister, W. Holmes, A. T. Lee, and P. L. Richards, Voltage-Biased Superconducting Transition-Edge Bolometer with Strong Electrothermal Feedback Operated at 370 mK, Applied Optics 37, 3391 (1998)

work page 1998

[3] [3]

K. D. Irwin, An application of electrothermal feedback for high resolution cryogenic particle detection, Applied Physics Letters 66, 1998 (1995)

work page 1998

[4] [4]

de Wit, L

M. de Wit, L. Gottardi, E. Taralli, K. Nagayoshi, M. Ridder, H. Akamatsu, M. Bruijn, R. Hoogeveen, J. van der Kuur, K. Ravensberg, D. Vaccaro, J.-R. Gao, and J.-W. den Herder, Impact of the Absorber- Coupling Design for Transition-Edge-Sensor X-Ray Calorimeters, Physical Review Applied 16, 044059 6 (2021), publisher: American Physical Society

work page 2021

[5] [5]

Irwin and G

K. Irwin and G. Hilton, Transition-Edge Sensors, in Cryogenic Particle Detection , edited by C. Enss (Springer, Berlin, Heidelberg, 2005) pp. 63–150

work page 2005

[6] [6]

M. A. Dobbs, M. Lueker, K. A. Aird, A. N. Bender, B. A. Benson, L. E. Bleem, J. E. Carlstrom, C. L. Chang, H.-M. Cho, J. Clarke, T. M. Crawford, A. T. Crites, D. I. Flanigan, T. de Haan, E. M. George, N. W. Halverson, W. L. Holzapfel, J. D. Hrubes, B. R. Johnson, J. Joseph, R. Keisler, J. Kennedy, Z. Kermish, T. M. Lanting, A. T. Lee, E. M. Leitch, D. Luo...

work page 2012

[7] [7]

T. de Haan, MNTES: modeling nonlinearity of TES detectors for enhanced cosmic microwave background measurements with LiteBIRD, in Millimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for Astronomy XII , Vol. 13102, edited by J. Zmuidzinas and J.-R. Gao (SPIE, 2024) p. 1310208, backup Publisher: International Society for Optics and Photonics

work page 2024

[8] [8]

Y. Zhou, T. de Haan, H. Akamatsu, D. Kaneko, M. Hazumi, M. Hasegawa, A. Suzuki, and A. T. Lee, A Method of Measuring TES Complex ETF Response in Frequency-Domain Multiplexed Readout by Single Sideband Power Modulation, Journal of Low Temperature Physics 10.1007/s10909-024-03107-z (2024)

work page doi:10.1007/s10909-024-03107-z 2024