Oscillators in Action
Temperature compensation for VCO
Temperature compensation for the VCO oscillation frequency is a critical issue
TODO π
Perturbation Projection Vector (PPV)
TODO π
Helene Thibieroz, Customer Support CIC. Using Spectre RF Noise-Aware PLL Methodology to Predict PLL Behavior Accurately [https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=3056e59ea76165373f90152f915a829d25dabebc]
Aditya Varma Muppala. Perturbation Projection Vector (PPV) Theory | Oscillators 11 | MMIC 16 [https://youtu.be/rbbHfZGT6Jo]
Limit Cycles
Nonlinear Dynamics
[https://adityamuppala.github.io/assets/Notes_YouTube/MMIC_Limit_Cycles.pdf]
ISF assumption
[https://adityamuppala.github.io/assets/Notes_YouTube/Oscillators_ISF_model.pdf]
Periodic ISF: Noise Folding
When performing the phase noise computation integral, there will be a negligible contribution from all terms, other than \(n=m\)
Given \(i(t) = I_m \cos[(m\omega_0 +\Delta \omega)t]\),
\[\begin{align} \phi(t) &= \frac{1}{q_\text{max}}\left[\frac{C_0}{2}\int_{-\infty}^t I_m\cos((m\omega_0 +\Delta \omega)\tau)d\tau + \sum_{n=1}^\infty C_n\int_{-\infty}^t I_m\cos((m\omega_0 +\Delta \omega)\tau)\cos(n\omega_0\tau)d\tau\right] \\ &= \frac{I_m}{q_\text{max}}\left[\frac{C_0}{2}\int_{-\infty}^t \cos((m\omega_0 +\Delta \omega)\tau)d\tau + \sum_{n=1}^\infty C_n\int_{-\infty}^t \frac{\cos((m\omega_0 + \Delta \omega+ n\omega_0)\tau)+ \cos((m\omega_0+\Delta \omega - n\omega_0)\tau)}{2}d\tau\right] \end{align}\]
If \(m=0\) \[ \phi(t) \approx \frac{I_0C_0}{2q_\text{max}\Delta \omega}\sin(\Delta\omega t) \] If \(m\neq 0\) and \(m=n\) \[ \phi(t) \approx \frac{I_mC_m}{2q_\text{max}\Delta \omega}\sin(\Delta\omega t) \]
\(m\omega_0 +\Delta \omega \ge 0\)
A. Hajimiri and T. H. Lee, "A general theory of phase noise in electrical oscillators," in IEEE Journal of Solid-State Circuits, vol. 33, no. 2, pp. 179-194, Feb. 1998
Corrections to "A General Theory of Phase Noise in Electrical Oscillators"
A. Hajimiri and T. H. Lee, "Corrections to "A General Theory of Phase Noise in Electrical Oscillators"," in IEEE Journal of Solid-State Circuits, vol. 33, no. 6, pp. 928-928, June 1998 [https://sci-hub.se/10.1109/4.678662]
Ali Hajimiri. Phase Noise in Oscillators [http://www-smirc.stanford.edu/papers/Orals98s-ali.pdf]
L. Lu, Z. Tang, P. Andreani, A. Mazzanti and A. Hajimiri, "Comments on βComments on βA General Theory of Phase Noise in Electrical Oscillatorsββ," in IEEE Journal of Solid-State Circuits, vol. 43, no. 9, pp. 2170-2170, Sept. 2008 [https://sci-hub.se/10.1109/JSSC.2008.2005028]
Given \(i(t) = I_m \cos[(m\omega_0 - \Delta \omega)t]\) and \(m \ge 1\)
\[\begin{align} \phi(t) &= \frac{1}{q_\text{max}}\left[\frac{C_0}{2}\int_{-\infty}^t I_m\cos((m\omega_0 -\Delta \omega)\tau)d\tau + \sum_{n=1}^\infty C_n\int_{-\infty}^t I_m\cos((m\omega_0 -\Delta \omega)\tau)\cos(n\omega_0\tau)d\tau\right] \\ &= \frac{I_m}{q_\text{max}}\left[\frac{C_0}{2}\int_{-\infty}^t \cos((m\omega_0 -\Delta \omega)\tau)d\tau + \sum_{n=1}^\infty C_n\int_{-\infty}^t \frac{\cos((m\omega_0 - \Delta \omega+ n\omega_0)\tau)+ \cos((m\omega_0-\Delta \omega - n\omega_0)\tau)}{2}d\tau\right] \end{align}\]
If \(m\ge 1\) and \(m=n\) \[ \phi(t) \approx \frac{I_mC_m}{2q_\text{max}\Delta \omega}\sin(\Delta\omega t) \] That is
| \(m = 0\) | \(m\gt 0\) & \(m\omega_0+\Delta \omega\) | \(m\gt 0\) & \(m\omega_0-\Delta \omega\) | |
|---|---|---|---|
| \(\phi(t)\) | \(\frac{I_0C_0}{2q_\text{max}\Delta \omega}\sin(\Delta\omega t)\) | \(\frac{I_mC_m}{2q_\text{max}\Delta \omega}\sin(\Delta\omega t)\) | \(\frac{I_mC_m}{2q_\text{max}\Delta \omega}\sin(\Delta\omega t)\) |
| \(P_{SBC}(\Delta \omega)\) | \(10\log(\frac{I_0^2C_0^2}{16q_\text{max}^2\Delta \omega^2})\) | \(10\log(\frac{I_m^2C_m^2}{16q_\text{max}^2\Delta \omega^2})\) | \(10\log(\frac{I_m^2C_m^2}{16q_\text{max}^2\Delta \omega^2})\) |
\[\begin{align} \mathcal{L}\{\Delta \omega\} &= 10\log\left(\frac{I_0^2C_0^2}{16q_\text{max}^2\Delta \omega^2} + 2\frac{I_m^2C_m^2}{16q_\text{max}^2\Delta \omega^2}\right) \\ &= 10\log\left(\frac{\overline{i_n^2/\Delta f}\cdot \frac{C_0^2}{2} }{4q_\text{max}^2\Delta \omega^2} + \frac{\overline{i_n^2/\Delta f}\cdot\sum_{m=1}^\infty C_m^2 }{4q_\text{max}^2\Delta \omega^2}\right) \\ &= 10\log \frac{\overline{i_n^2/\Delta f}(C_0^2/2+\sum_{m=1}^\infty C_m^2)}{4q_\text{max}^2\Delta \omega^2} \\ &= 10\log \frac{\overline{i_n^2/\Delta f}\cdot \Gamma_\text{rms}^2}{2q_\text{max}^2\Delta \omega^2} \end{align}\]
Carlo Samori, Phase Noise in LC Oscillators: From Basic Concepts to Advanced Topologies [https://www.ieeetoronto.ca/wp-content/uploads/2020/06/DL-VCO-short.pdf]
ISF & \(1/f\)-noise up-conversion
TODO π
ISF Simulation
PSS + PXF Method
Yizhe Hu, "A Simulation Technique of Impulse Sensitivity Function (ISF) Based on Periodic Transfer Function (PXF)" [https://bbs.eetop.cn/thread-869343-1-1.html]
TODO π
Transient Method
David Dolt. ECEN 620 Network Theory - Broadband Circuit Design: "VCO ISF Simulation" [https://people.engr.tamu.edu/spalermo/ecen620/ISF_SIM.pdf]
To compare the ring oscillator and VCO the total injected charge to both should be the same
Tail filter
TODO π
P. Liu et al., "A 128Gb/s ADC/DAC Based PAM-4 Transceiver with >45dB Reach in 3nm FinFET," 2025 Symposium on VLSI Technology and Circuits (VLSI Technology and Circuits), Kyoto, Japan, 2025
E. Hegazi, H. Sjoland and A. Abidi, "A filtering technique to lower oscillator phase noise," 2001 IEEE International Solid-State Circuits Conference. Digest of Technical Papers. ISSCC (Cat. No.01CH37177), San Francisco, CA, USA, 2001 [paper, slides]
β, "A filtering technique to lower LC oscillator phase noise," in IEEE Journal of Solid-State Circuits, vol. 36, no. 12, pp. 1921-1930, Dec. 2001 [https://people.engr.tamu.edu/spalermo/ecen620/filtering_tech_lc_osc_hegazi_jssc_2001.pdf]
D. Murphy, H. Darabi and H. Wu, "Implicit Common-Mode Resonance in LC Oscillators," in IEEE Journal of Solid-State Circuits, vol. 52, no. 3, pp. 812-821, March 2017, [https://sci-hub.st/10.1109/JSSC.2016.2642207]
β, "25.3 A VCO with implicit common-mode resonance," 2015 IEEE International Solid-State Circuits Conference - (ISSCC) Digest of Technical Papers, San Francisco, CA, USA, 2015 [https://sci-hub.st/10.1109/ISSCC.2015.7063116]
Lecture 16: VCO Phase Noise [https://people.engr.tamu.edu/spalermo/ecen620/lecture16_ee620_vco_pn.pdf]
Injection Locking
TODO π
"Topics in IC (Wireline Transceiver Design): Lec 4 - Injection Locked Oscillators" [https://ocw.snu.ac.kr/sites/default/files/NOTE/Lec%204%20-%20Injection%20Locked%20Oscillators.pdf]
Cowan, Glenn. (2024). Mixed-Signal CMOS for Wireline Communication: Transistor-Level and System-Level Design Considerations
Mozhgan Mansuri. ISSCC2021 SC3: Clocking, Clock Distribution, and Clock Management in Wireline/Wireless Subsystems [https://www.nishanchettri.com/isscc-slides/2021%20ISSCC/SHORT%20COURSE/ISSCC2021-SC3.pdf]
Aditya Varma Muppala. Oscillator Theory - Injection Locking [note, video1, video2]
Min-Seong Choo. Review of Injection-Locked Oscillators [https://journal.theise.org/jse/wp-content/uploads/sites/2/2020/09/JSE-2020-0001.pdf]
capacitance simulation
VCO varactor
Two methods: 1. pss + pac; 2. pss+psp
PSS + PAC
pss time domain
using the 0-harmonic
PSS + PSP
using Y11 of psp
comparison
which are same
inverter input
R-C, series equivalent circuit
inverter output
R-C, parallel equivalent circuit
AC simulation
@vi = 0
sweep vi from 0 to 800mV (vdd)
SP simulation
EEStream. Cadence - How to find device capacitance - DC simulation, SP simulation and Large-signal SP simulation [https://www.youtube.com/watch?v=M3zP6eJnONk]
reference
JiΕΓ Lebl. Notes on Diffy Qs: Differential Equations for Engineers [link]
Matt Charnley. Differential Equations: An Introduction for Engineers [link]
Γ strΓΆm, 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]
Cadence Blog, "Resonant Frequency vs. Natural Frequency in Oscillator Circuits" [link]
Aditya Varma Muppala. Oscillators [https://youtube.com/playlist?list=PL9Trid0A4Da2fOmYTEjhAnUkGPxyiH7H6&si=ILxn8hfkMYjXW5f4]
P.E. Allen - 2003. ECE 6440 - Frequency Synthesizers: Lecture 160 β Phase Noise - II [https://pallen.ece.gatech.edu/Academic/ECE_6440/Summer_2003/L160-PhNoII(2UP).pdf]
Y. Hu, T. Siriburanon and R. B. Staszewski, "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]
S. Levantino, P. Maffezzoni, F. Pepe, A. Bonfanti, C. Samori and A. L. Lacaita, "Efficient Calculation of the Impulse Sensitivity Function in Oscillators," in IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 59, no. 10, pp. 628-632, Oct. 2012 [https://sci-hub.se/10.1109/TCSII.2012.2208679]
S. Levantino and P. Maffezzoni, "Computing the Perturbation Projection Vector of Oscillators via Frequency Domain Analysis," in IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, vol. 31, no. 10, pp. 1499-1507, Oct. 2012 [https://sci-hub.se/10.1109/TCAD.2012.2194493]
Thomas H. Lee. Linearity, Time-Variation, Phase Modulation and Oscillator Phase Noise [https://class.ece.iastate.edu/djchen/ee507/PhaseNoiseTutorialLee.pdf]
Y. Hu, T. Siriburanon and R. B. Staszewski, "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 [https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=9286468]
Jaeha Kim. Lecture 8. Special Topics: Design Trade -Offs in LC -Tuned Oscillators [https://ocw.snu.ac.kr/sites/default/files/NOTE/7033.pdf]
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.se/10.1109/81.847872]
A. A. Abidi and D. Murphy, "How to Design a Differential CMOS LC Oscillator," in IEEE Open Journal of the Solid-State Circuits Society, vol. 5, pp. 45-59, 2025 [https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=10818782]
Akihide Sai, Toshiba. ISSCC 2023 T5: All-digital PLLs From Fundamental Concepts to Future Trends [https://www.nishanchettri.com/isscc-slides/2023%20ISSCC/TUTORIALS/T5.pdf]
Pietro Andreani. ISSCC 2011 T1: Integrated LC oscillators [slides,transcript]
β. ISSCC 2017 F2: Integrated Harmonic Oscillators
β. SSCS Distinguished Lecture: RF Harmonic Oscillators Integrated in Silicon Technologies [https://www.ieeetoronto.ca/wp-content/uploads/2020/06/DL-Toronto.pdf]
β. ESSCIRC 2019 Tutorials: RF Harmonic Oscillators Integrated in Silicon Technologies [https://youtu.be/k1I9nP9eEHE?si=fns9mf3aHjMJobPH]
Jun Yin. ISSCC 2025 T10: mm-Wave Oscillator Design