Noise in Nonlinear Circuits And Systems

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flicker noise upconversion

Y. Hu, T. Siriburanon and R. B. Staszewski, "A Low-Flicker-Noise 30-GHz Class-F23 Oscillator in 28-nm CMOS Using Implicit Resonance and Explicit Common-Mode Return Path," in IEEE Journal of Solid-State Circuits, vol. 53, no. 7, pp. 1977-1987, July 2018 [https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=8345650]

—, "Intuitive Understanding of Flicker Noise Reduction via Narrowing of Conduction Angle in Voltage-Biased Oscillators," in IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 66, no. 12, pp. 1962-1966, Dec. 2019 [https://sci-hub.se/10.1109/TCSII.2019.2896483]

—, "Oscillator Flicker Phase Noise: A Tutorial," in IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 68, no. 2, pp. 538-544, Feb. 2021 [paper] [slides]

E. G. Ioannidis, C. G. Theodorou, T. A. Karatsori, S. Haendler, C. A. Dimitriadis and G. Ghibaudo, "Drain-Current Flicker Noise Modeling in nMOSFETs From a 14-nm FDSOI Technology," in IEEE Transactions on Electron Devices, vol. 62, no. 5, pp. 1574-1579, May 2015 [https://sci-hub.jp/10.1109/TED.2015.2411678]

Flicker Noise Modulation

Flicker Noise Formulations in Compact Models

G. J. Coram, C. C. McAndrew, K. K. Gullapalli and K. S. Kundert, "Flicker Noise Formulations in Compact Models," in IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, vol. 39, no. 10, pp. 2812-2821, Oct. 2020 [https://kenkundert.com/docs/tcad20-flicker-noise.pdf],[https://github.com/KenKundert/flicker-noise]

BSIM4v4.7 MOSFET Model -User's Manual [https://class.ece.iastate.edu/djchen/ee501/BSIM470_Manual.pdf]

flicker noise in circuit-noise analysis

its power spectral density is approximately \[ S_{i,1/f}(f)=\frac{K}{|f|}. \] A large amount of its power lies at low frequencies. Therefore, compared with a GHz oscillation period \(T_0\), the flicker-noise value changes very little during one cycle.

For a flicker-noise component at frequency \(f_m\), \[ f_m T_0\ll 1 \] implies \[ i_{1/f}(t+T_0)\approx i_{1/f}(t). \] Thus, if the noise current is positive at \(t_0\), it will probably remain positive throughout the following oscillator cycle: \[ i_{1/f}(t_0+\tau)\approx i_{1/f}(t_0), \qquad 0\leq \tau<T_0. \] In circuit-noise analysis, the underlying flicker-noise source is commonly treated as approximately wide-sense stationary: \[ R_x(t_1,t_2)=R_x(t_1-t_2). \] This is reasonable when the device bias is constant and the measurement interval is finite.

The phase perturbations may cancel or leave a nonzero residual: \[ \Delta\phi_{\text{cycle}} \propto \int_{0}^{T_0} \Gamma(\omega_0 t)\, i_{1/f,\mathrm{cyclo}}(t)\,dt. \] Since the low-frequency noise is almost constant over \(T_0\), \[ \Delta\phi_{\text{cycle}} \approx x_{1/f}(t_0) \int_{0}^{T_0} \Gamma(\omega_0 t)a(t)\,dt \] Therefore, flicker-noise upconversion depends on whether the phase-delay and phase-advance contributions cancel over one period. A nonzero weighted average produces low-frequency fluctuations in oscillator frequency, which commonly appear as the \(1/f^3\) phase-noise region.

Define

\[ \Gamma_{\mathrm{eff,DC}}\equiv \frac{1}{T_0}\int_0^{T_0}\Gamma(\omega_0t)a(t)\,dt \]

Then \[ \Delta\phi_{\text{cycle}} \approx \frac{x_{1/f}(t_0)}{q_{\max}} \Gamma_{\mathrm{eff,DC}}T_0. \] If \(x_{1/f}\) is already normalized by \(q_{\max}\), the \(1/q_{\max}\) factor can be omitted.

Therefore, \[ \boxed{\Gamma_{\mathrm{eff,DC}}=0 \quad\Longrightarrow\quad \Delta\phi_{\text{cycle}}\approx 0} \] for quasistatic flicker noise. Physically, the phase-delay contribution on one edge exactly cancels the phase-advance contribution on the other edge.nce, \[ \boxed{ \Gamma_{\mathrm{eff,DC}}=0 \Rightarrow \text{no first-order direct }1/f\text{-to-}1/f^3 \text{ phase-noise upconversion from that source.} } \]

Mathematical Preliminaries

Jiří Lebl. Notes on Diffy Qs: Differential Equations for Engineers [link]

Matt Charnley. Differential Equations: An Introduction for Engineers [link]

Åström, K.J. & Murray, Richard. (2021). Feedback Systems: An Introduction for Scientists and Engineers Second Edition [https://www.cds.caltech.edu/~murray/books/AM08/pdf/fbs-public_24Jul2020.pdf]

Strogatz, S.H. (2015). Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (2nd ed.). CRC Press [https://www.biodyn.ro/course/literatura/Nonlinear_Dynamics_and_Chaos_2018_Steven_H._Strogatz.pdf]

Godone, A. & Micalizio, Salvatore & Levi, Filippo. (2008). RF spectrum of a carrier with a random phase modulation of arbitrary slope. [https://sci-hub.se/10.1088/0026-1394/45/3/008]

reference

A. Demir, A. Mehrotra and J. Roychowdhury, "Phase noise in oscillators: a unifying theory and numerical methods for characterization," in IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 47, no. 5, pp. 655-674, May 2000 [https://sci-hub.jp/10.1109/81.847872]

A. Demir and A. Sangiovanni-Vincentelli, Analysis and Simulation of Noise in Nonlinear Electronic Circuits and Systems, vol. 425. Boston, MA, USA: Kluwer Academic Publishers, 1998

A. Mehrotra and A. Sangiovanni-Vincentelli, Noise Analysis of Radio Frequency Circuits, 1st ed. New York, NY, USA: Springer, 2004

Darabi H. Radio Frequency Integrated Circuits and Systems. 2nd ed. Cambridge University Press; 2020.