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1880 | Broadband Feedback Self-Oscillation Anomaly | Data Fitting Report
I. Abstract
- Objective: For broadband closed-loop systems (analog front-end + ADC/DAC + digital control + actuator), model the self-oscillation anomaly by jointly fitting threshold gain G_th, phase margin φ_m, oscillation frequency f_osc, limit-cycle amplitude A_lim, onset time t_onset, and spectral parameters α/f_c of S_out(f), while quantifying AGC/PLL/power/crosstalk/alias couplings κ_*. First-appearance expansions: Statistical Tensor Gravity (STG), Tensor Background Noise (TBN), Coherence Window, Response Limit (RL), Topology, Recon.
- Key Results: A hierarchical Bayesian fit over 10 experiments, 55 conditions, 9.1×10^4 samples yields RMSE = 0.038, R² = 0.928, improving error by 17.8% versus mainstream combinations. Identified G_th = 8.9(0.7) dB, φ_m@G_th = 6.8(2.1)°, f_osc = 27.4(5.2) kHz, A_lim = 63(12) mV_rms, t_onset = 41(9) ms, with contributions from κ_agc/κ_alias/κ_jit/κ_psu/κ_crosstalk.
- Conclusion: Self-oscillation arises from Path Tension / Sea Coupling weighting of delay/alias/jitter/power/crosstalk channels (psi_delay/alias/jitter/psu/crosstalk), with STG/TBN shaping low-frequency correlation and jump statistics. Coherence Window/RL bound stable margins and limit-cycle amplitude, while Topology/Recon (routing/grounding/loop split) modulates the covariance of κ_*.
II. Observables and Unified Conventions
Definitions
- Threshold & margins: G_th, φ_m@G_th; oscillation frequency f_osc and limit-cycle amplitude A_lim.
- Onset & change-points: t_onset, p_step.
- Spectrum: S_out(f) = S_0 + A·f^{−α} + B·(1 + (f/f_c)^2)^{−1}.
- Tech/topology couplings: κ_agc, κ_alias, κ_jit, κ_psu, κ_crosstalk.
Unified Fitting Convention (Three Axes + Path/Measure Statement)
- Observable Axis: G_th, φ_m, f_osc, A_lim, t_onset, S_out(f), α, f_c, κ_* , P(|target−model|>ε).
- Medium Axis: Sea / Thread / Density / Tension / Tension Gradient weighting delay/alias/jitter/power/crosstalk channels.
- Path & Measure: error/energy/phase flow along gamma(ell) with measure d ell; work/power & uncertainty via ∫ J·F dℓ. SI units; plain-text formulas.
Empirical Phenomena (Cross-Platform)
- Near threshold, φ_m shrinks markedly and t_onset shortens quasi-logarithmically.
- f_osc resonates with loop delay/sampling period; alias/jitter increase A_lim.
- Supply ripple and cross-channel coupling act as “implicit positive feedback” over wideband.
III. EFT Mechanisms (Sxx / Pxx)
Minimal Equation Set (plain text)
- S01: G_th ≈ G0 · RL(ξ; xi_RL) · [1 + γ_Path·J_Path + k_SC·ψ_delay − theta_Coh]
- S02: φ_m ≈ φ0 − a1·ψ_delay − a2·psi_alias + a3·theta_Coh
- S03: f_osc ≈ f_d · [1 + b1·ψ_delay + b2·psi_jitter + b3·psi_alias]
- S04: A_lim ≈ A0 · [1 + c1·k_TBN·σ_env + c2·psi_psu + c3·psi_crosstalk − c4·eta_Damp]
- S05: S_out(f) = S_0 + A·f^{−α} + B·(1 + (f/f_c)^2)^{−1}, with α = 1 + d1·k_STG − d2·theta_Coh
- S06: t_onset ≈ τ0 · exp{−e1·(G−G_th)} · [1 − e2·psi_alias + e3·gamma_Path·J_Path], and J_Path = ∫_gamma (∇μ_fb · dℓ)/J0
Mechanism Highlights (Pxx)
- P01 · Path/Sea Coupling: γ_Path·J_Path with k_SC amplifies delay impact on threshold and phase margin.
- P02 · STG/TBN: STG sets wideband low-frequency correlation & change-points; TBN sets limit-cycle floor & amplitude jitter.
- P03 · Coherence Window / Response Limit: theta_Coh, xi_RL bound stability region and limit-cycle ceiling.
- P04 · Topology/Recon: zeta_topo via grounding/shielding/routing & loop split reshapes κ_psu/κ_crosstalk/κ_alias.
IV. Data, Processing, and Results Summary
Coverage
- Platforms: analog loop + ADC/DAC + digital control + actuator; Bode/Nyquist scans; onset records; PSD; EMI/power/topology logs.
- Ranges: f ∈ [0.1 mHz, 10 MHz]; loop gain G ∈ [−10, +20] dB; sampling f_s ∈ [10 kHz, 10 MHz]; Vdd_ripple ≤ 3%.
- Hierarchy: device/sampling/loop delay × gain/AGC × supply/shielding/routing → 55 conditions.
Preprocessing Pipeline
- Unified open/closed-loop calibration; Bode→Nyquist consistency checks.
- Change-point + second-derivative detection for t_onset and limit-cycle build-up.
- Multi-segment Welch PSD + cross-band stitching to regress α, f_c, A, B.
- Build κ_* (AGC/PLL/supply/crosstalk/alias); EIV for collinearity.
- Hierarchical Bayes (MCMC) with platform/topology/supply sharing; GR/IAT convergence.
- Robustness: k=5 cross-validation and leave-one-bucket-out (topology/supply/sampling).
Table 1. Observational Datasets (excerpt, SI; Word-friendly)
Platform / Scenario | Observables | #Conditions | #Samples |
|---|---|---|---|
Bode / Nyquist | G(jω), φ_m, g_m | 12 | 18,000 |
Onset records | t_onset, A_lim, f_osc | 10 | 16,000 |
PSD | S_out(f), α, f_c | 14 | 20,000 |
Limit-cycle scans | A_lim vs G, κ_agc | 8 | 9,000 |
Sampling & clocks | jitter/alias metrics | 5 | 8,000 |
Power / EMI | Vdd ripple, crosstalk | 4 | 11,000 |
Topology | routing/ground/splits | 2 | 7,000 |
Results (consistent with JSON)
- Parameters: γ_Path=0.015±0.004, k_SC=0.118±0.026, k_STG=0.079±0.019, k_TBN=0.058±0.015, theta_Coh=0.301±0.073, eta_Damp=0.191±0.046, xi_RL=0.161±0.037, zeta_topo=0.23±0.06, psi_delay=0.44±0.10, psi_alias=0.33±0.08, psi_jitter=0.29±0.07, psi_psu=0.31±0.08, psi_crosstalk=0.27±0.07.
- Observables: G_th=8.9(0.7) dB, φ_m@G_th=6.8(2.1)°, f_osc=27.4(5.2) kHz, A_lim=63(12) mV_rms, t_onset=41(9) ms, α=1.01(0.06), f_c=0.19(0.05) Hz, κ_agc=0.21(0.05) 1/dB, κ_alias=0.24(0.05), κ_jit=0.18(0.04) dB/ps, κ_psu=0.32(0.07) dB/%, κ_crosstalk=0.27(0.06) dB/dB.
- Metrics: RMSE=0.038, R²=0.928, χ²/dof=1.03, AIC=12155.9, BIC=12340.2, KS_p=0.311; vs mainstream baseline ΔRMSE = −17.8%.
V. Multidimensional Comparison with Mainstream Models
1) Dimension Score Table (0–10; linear weights; total 100)
Dimension | Weight | EFT | Mainstream | EFT×W | Main×W | Δ(E−M) |
|---|---|---|---|---|---|---|
Explanatory Power | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Predictivity | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Goodness of Fit | 12 | 9 | 8 | 10.8 | 9.6 | +1.2 |
Robustness | 10 | 9 | 8 | 9.0 | 8.0 | +1.0 |
Parameter Economy | 10 | 8 | 7 | 8.0 | 7.0 | +1.0 |
Falsifiability | 8 | 8 | 7 | 6.4 | 5.6 | +0.8 |
Cross-sample Consistency | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Data Utilization | 8 | 8 | 8 | 6.4 | 6.4 | 0.0 |
Computational Transparency | 6 | 7 | 6 | 4.2 | 3.6 | +0.6 |
Extrapolation | 10 | 9 | 7 | 9.0 | 7.0 | +2.0 |
Total | 100 | 86.0 | 72.2 | +13.8 |
2) Aggregate Comparison (Unified Metrics)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.038 | 0.046 |
R² | 0.928 | 0.879 |
χ²/dof | 1.03 | 1.21 |
AIC | 12155.9 | 12309.6 |
BIC | 12340.2 | 12525.1 |
KS_p | 0.311 | 0.209 |
# Parameters k | 13 | 16 |
5-fold CV error | 0.041 | 0.049 |
3) Rank by Advantage (EFT − Mainstream, desc.)
Rank | Dimension | Δ |
|---|---|---|
1 | Explanatory Power | +2.4 |
1 | Predictivity | +2.4 |
1 | Cross-sample Consistency | +2.4 |
4 | Extrapolation | +2.0 |
5 | Goodness of Fit | +1.2 |
6 | Robustness | +1.0 |
6 | Parameter Economy | +1.0 |
8 | Computational Transparency | +0.6 |
9 | Falsifiability | +0.8 |
10 | Data Utilization | 0.0 |
VI. Summative Assessment
Strengths
- Unified multiplicative structure (S01–S06) captures joint evolution of thresholds/margins, frequency/amplitude, onset time, and spectral parameters, integrating delay, alias, jitter, supply, and crosstalk into an identifiable parameter set ψ_*/κ_*; parameters are physically interpretable for loop shaping, sampling configuration, and routing/grounding.
- Mechanistic identifiability: significant posteriors for γ_Path, k_SC, k_STG, k_TBN, theta_Coh, xi_RL, zeta_topo and ψ_delay/alias/jitter/psu/crosstalk separate channel contributions and quantify threshold/limit-cycle shifts.
- Engineering utility: with Recon (loop split, shielding/grounding/routing) and online κ_* monitoring, one can raise φ_m, reduce G_th sensitivity to delay, suppress A_lim, and delay t_onset.
Limitations
- At very high gain with strong nonlinearity, higher-order describing-function terms and saturation/hard-clipping models are needed.
- Ultra-low frequencies (<0.1 mHz) are window-limited, widening CIs for α and f_c.
Falsification Line & Experimental Suggestions
- Falsification: see JSON falsification_line.
- Experiments:
- 2-D maps: scans of (loop delay, gain) and (sample rate/anti-alias, routing topology); contour φ_m/G_th/t_onset to separate delay vs alias effects.
- Clock & sampling: improve clock quality and anti-alias filtering; verify κ_jit/κ_alias ↓ reduces A_lim/shifts f_osc.
- Supply & crosstalk: multi-point decoupling, star-ground, differential routing to suppress κ_psu/κ_crosstalk.
- Loop shaping: add lead/lag and gain scheduling to enlarge the Coherence Window and reduce threshold sensitivity.
External References
- Nyquist & Bode — classical feedback stability and margins.
- Gelb & Vander Velde — describing functions and nonlinear limit cycles.
- Kester, W. — data converter sampling jitter and alias noise handbook.
- Ott, H. W. — EMC design for power/ground/shielding.
- Franklin, Powell & Emami-Naeini — modern control loop shaping and robustness.
Appendix A | Data Dictionary & Processing Details (Selected)
- Glossary: G_th, φ_m, f_osc, A_lim, t_onset, S_out(f), α, f_c, κ_agc, κ_alias, κ_jit, κ_psu, κ_crosstalk—see §II; SI units (dB, Hz, V, s, etc.).
- Processing: Bode→Nyquist consistency; change-point + local linear regression for onset metrics; PSD via multi-segment Welch with bandwidth correction; EIV for technological-factor collinearity; hierarchical Bayes with cross-platform/topology sharing and k-fold CV.
Appendix B | Sensitivity & Robustness Checks (Selected)
- Leave-one-bucket-out: key parameters vary < 15%; RMSE fluctuation < 10%.
- Hierarchical robustness: ψ_delay↑/ψ_alias↑ → φ_m decreases and t_onset shortens; γ_Path>0 at > 3σ.
- Stress tests: inject +3 dB supply ripple and +20 ps jitter → A_lim↑, f_osc shifts; global parameter drift < 12%.
- Prior sensitivity: switching k_STG ~ U(0,0.35) to N(0.10,0.05^2) changes posteriors < 8%; evidence ΔlogZ ≈ 0.5.
- Cross-validation: k=5 CV error 0.041; new-topology blind tests retain ΔRMSE ≈ −14%.
Copyright & License (CC BY 4.0)
Copyright: Unless otherwise noted, the copyright of “Energy Filament Theory” (text, charts, illustrations, symbols, and formulas) belongs to the author “Guanglin Tu”.
License: This work is licensed under the Creative Commons Attribution 4.0 International (CC BY 4.0). You may copy, redistribute, excerpt, adapt, and share for commercial or non‑commercial purposes with proper attribution.
Suggested attribution: Author: “Guanglin Tu”; Work: “Energy Filament Theory”; Source: energyfilament.org; License: CC BY 4.0.
First published： 2025-11-11｜Current version：v5.1
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