LC Oscillator

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LC Oscillator Structures

Class-C CMOS Oscillator

A. Mazzanti and P. Andreani, "Class-C Harmonic CMOS VCOs, With a General Result on Phase Noise," in IEEE Journal of Solid-State Circuits, vol. 43, no. 12, pp. 2716-2729, Dec. 2008 [https://sci-hub.ru/10.1109/JSSC.2008.2004867]

TODO 📅

Class-D CMOS Oscillator

L. Fanori and P. Andreani, "A 2.5-to-3.3GHz CMOS Class-D VCO," 2013 IEEE International Solid-State Circuits Conference Digest of Technical Papers, San Francisco, CA, USA, 2013 [https://sci-hub.red/10.1109/ISSCC.2013.6487763]

—, "Class-D CMOS Oscillators," in IEEE Journal of Solid-State Circuits, vol. 48, no. 12, pp. 3105-3119, Dec. 2013 [https://sci-hub.red/10.1109/JSSC.2013.2271531]

—, "A Class-D CMOS DCO with an on-chip LDO," ESSCIRC 2014 - 40th European Solid State Circuits Conference (ESSCIRC), Venice Lido, Italy, 2014 [https://sci-hub.red/10.1109/ESSCIRC.2014.6942090]

TODO 📅

Class-F CMOS Oscillator

Huijung Kim, Seonghan Ryu, Yujin Chung, Jinsung Choi and Bumman Kim, "A low phase-noise CMOS VCO with harmonic tuned LC tank," in IEEE Transactions on Microwave Theory and Techniques, vol. 54, no. 7, pp. 2917-2924, July 2006 [https://sci-hub.ru/10.1109/tmtt.2006.877439]

M. Babaie and R. B. Staszewski, "Third-harmonic injection technique applied to a 5.87-to-7.56GHz 65nm CMOS Class-F oscillator with 192dBc/Hz FOM," 2013 IEEE International Solid-State Circuits Conference Digest of Technical Papers, San Francisco, CA, USA, 2013 [https://sci-hub.ru/10.1109/ISSCC.2013.6487764]

—, "A Class-F CMOS Oscillator," in IEEE Journal of Solid-State Circuits, vol. 48, no. 12, pp. 3120-3133, Dec. 2013 [https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6576263]

TODO 📅

conduction angle

TODO 📅

Class-B Oscillators

Output Amplitude

Edgar Sanchez-Sinencio. ECEN 665, OSCILLATORS [https://people.engr.tamu.edu/s-sanchez/665%20Oscillators.pdf]

NMOS Realization — single pair

image-20260622230057948

common mode current don't contribute to output amplitude

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L0 = 1e-9 * 2;
RL0 = 0.25133 * 2;
C0 = 6.333e-12 / 2;
RC0 = 0.50264 * 2;

w0 = 1/sqrt(L0*C0);   % 12.566 Grad/s

QL = w0*L0/RL0;       % 50
QC = 1/(w0*C0)/RC0;   % 25

RLp0 = QL^2 * RL0;
RCp0 = QC^2 * RC0;
Rp = RLp0 * RCp0 / (RLp0 + RCp0);  % 418.8576 Ohm
Qtot_by_L = Rp/(w0*L0);    % 16.6664
Qtot_by_C = Rp*(w0*C0);    % 16.6664

I0 = 0.5e-3;
vp_p = 2/pi * I0 * Rp/2;     % 66.6633 mV

%%%% compute Qtot from simulation waveform
vp_p2p_sim = 132.8e-3;
Qtot_calc_L0 = vp_p2p_sim*pi/2/I0/(w0*L0);   % 16.6006
Qtot_calc_C0 = vp_p2p_sim*pi/2/I0*(w0*C0);   % 16.6006

CMOS Realization — double pair

image-20260622230206044

Owing to switch-off PMOS eliminating common mode current, all \(I_T\) is differentially flowing in the tank

image-20260622225133737

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current limited vs voltage limited

image-20260622205909171

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Class-B Power/Current Efficiency

Z. Wang, S. Diao, L. He, X. Jiang and F. Lin, "Analysis of Current Efficiency for CMOS Class-B LC Oscillators," in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 62, no. 5, pp. 1345-1352, May 2015 [https://sci-hub.jp/10.1109/TCSI.2015.2411792]

L. Bertulessi, S. Levantino and C. Samori, "Analysis of power efficiency in high-performance class-B oscillators," 2016 12th Conference on Ph.D. Research in Microelectronics and Electronics (PRIME), Lisbon, Portugal, 2016 [https://sci-hub.jp/10.1109/PRIME.2016.7519525]

TODO 📅

Current-biased & voltage-biased

S. Gallucci et al., "A Low-Noise Digital PLL With an Adaptive Common-Mode Resonance Tuning Technique for Voltage-Biased Oscillators," in IEEE Journal of Solid-State Circuits, vol. 60, no. 12, pp. 4572-4586, Dec. 2025

TODO 📅

image-20260106224228115

8-shaped inductor

NXP BV, US8183971B2, 8-shaped inductor [pdf]

Marvell, US9077310B2, Pseudo-8-shaped inductor [pdf]

P. Guan et al., "8-Shaped Inductors: An Essential Addition to RFIC Designers' Toolbox," in IEEE Open Journal of the Solid-State Circuits Society, vol. 4, pp. 131-146, 2024 [pdf]

M. Pisati et al., "A 243-mW 1.25–56-Gb/s Continuous Range PAM-4 42.5-dB IL ADC/DAC-Based Transceiver in 7-nm FinFET," in IEEE Journal of Solid-State Circuits, vol. 55, no. 1, pp. 6-18, Jan. 2020 [https://sci-hub.ru/10.1109/JSSC.2019.2936307]

An 8-shaped (figure-8) inductor is a specialized on-chip, high-Q component used to mitigate electromagnetic coupling and reduce frequency pulling in VCOs by generating opposing, self-canceling magnetic fields

TODO 📅

image-20260511195827682

Zou, Wei & Zou, Xuecheng & Ren, Daming & Zhang, Kefeng & Liu, Dongsheng & Ren, Zhixiong. (2019). 2.49-4.91 GHz wideband VCO with optimised 8-shaped inductor. Electronics Letters. [https://sci-hub.jp/10.1049/el.2018.6012]

image-20260511230451153

Capacitor Bank

B. Sadhu and R. Harjani, "Capacitor bank design for wide tuning range LC VCOs: 850MHz-7.1GHz (157%)," Proceedings of 2010 IEEE International Symposium on Circuits and Systems, Paris, France, 2010 [https://sci-hub.st/10.1109/ISCAS.2010.5537040]

TODO 📅

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large value KVCO is not favorable due to noise and possibly spurs at the control voltage

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LC Tank Q

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Definitions of Q

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Assuming RLC oscillator waveform is \(V(t)=V_0\sin\omega_0 t\), \(\omega_0 = \frac{1}{\sqrt{LC}}\) is resonant frequency

Energy stored \[ E_t = \frac{1}{2}LI_0^2 = \frac{1}{2}CV_0^2 \] Energy Dissipated per Cycle \[ E_d = \frac{V_0^2}{2R}\frac{2\pi}{\omega_0} \] For \(Q_4\), with \(I_0=C\omega_0V_0\) \[ \boxed{Q_4 = 2\pi\frac{E_s}{E_d} = R\omega_0C = \frac{R}{\omega_0L}} \]

which holds at resonance

image-20251012100816733

For \(Q_3\), suppose RLC tank is driven by \(V_o\cos \omega t\) voltage source, then

Peak Magnetic Energy \[ E_{pL} = \frac{1}{2}LI_0^2 = \frac{1}{2}L\left(\frac{V_0}{L\omega}\right)^2 \] Peak Electric Energy \[ E_{pC} = \frac{1}{2}CV_0^2 \] with Energy Lost per Cycle \(E_d = \frac{V_0^2}{2R}\frac{2\pi}{\omega_0}\), we have \[ Q_3 = \frac{E_{pL} - E_{pC}}{E_d} = \left(\frac{1}{L\omega^2}-C\right)R\omega=\frac{R}{L\omega}\left(1 - \frac{\omega^2}{\omega^2_{SR}}\right) \]

image-20251012100931217

EEE 211 ANALOG ELECTRONICS [https://www.ee.bilkent.edu.tr/~eee211/LectureNotes/Chapter%20-%2004.pdf]

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Tank Impedance Near Resonance

image-20260620150838359

image-20260620151219168 \[ \boxed{|Z(\omega_0\pm \omega_m)|^2 \cong \frac{1}{(2\omega_m C)^2}}\qquad\qquad \boxed{|Z(\omega_0\pm \omega_m)|^2 \cong R^2\left(\frac{\omega_0}{2Q\omega_m}\right)^2} \]

LC-VCO Temperature Sensitivities

A. L. S. Loke et al., "A versatile low-jitter PLL in 90-nm CMOS for SerDes transmitter clocking," Proceedings of the IEEE 2005 Custom Integrated Circuits Conference, 2005., San Jose, CA, USA, 2005 [slides, paper]

image-20251213154802429

\[ f=\frac{1}{2\pi\sqrt{L_p C_p}} = \frac{1}{2\pi\sqrt{L_s\frac{Q_L^2+1}{Q_L^2} C_s\frac{Q_C^2}{Q_C^2+1}}} = \frac{1}{2\pi\sqrt{L_sC_s}}\cdot \sqrt{\frac{1+1/Q_c^2}{1+1/Q_L^2}} \] Assuming the tank's Q is limited by the inductor's quality factor \(Q_L\), i.e. \(Q_L\ll Q_c\) \[ f\approx \frac{1}{2\pi\sqrt{L_sC_s}}\cdot \sqrt{1-\frac{1}{Q_L^2}} =f_0\cdot\sqrt{1-\frac{1}{Q_L^2}} \] where \(f_0=\frac{1}{\sqrt{L_sC_s}}\) is the first order approximation of the resonant frequency

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simulation in oscillator

varactor simulation

Three methods:

  • PSS +PSP (pay attention to port termination and voltage amplitude)
  • PSS +PAC
  • PSS Only

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rms only scale magnitude \(1/\sqrt{2}\) but retain phase for complex number, like harmonic

  • mag(vh('pss "/P5")) = mag(rms(vh('pss "/P5"))) * (2**0.5)
  • phaseDegUnwrapped(vh('pss "/P5")) = phaseDegUnwrapped(rms(vh('pss "/P5")))

image-20251026155120102

reference

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]

Pietro Andreani. ISSCC 2011 T1: Integrated LC oscillators

—. ISSCC 2017 F2: Integrated Harmonic Oscillators

—. SSCS Distinguished Lecture: RF Harmonic Oscillators Integrated in Silicon Technologies [https://www.ieeetoronto.ca/wp-content/uploads/2020/06/DL-Toronto.pdf]

—. ESSCIRC 2019 Tutorials: RF Harmonic Oscillators Integrated in Silicon Technologies [https://youtu.be/k1I9nP9eEHE]

—. "Harmonic Oscillators in CMOS—A Tutorial Overview," in IEEE Open Journal of the Solid-State Circuits Society, vol. 1, pp. 2-17, 2021 [pdf]

C. Samori, "Tutorial: Understanding Phase Noise in LC VCOs," 2016 IEEE International Solid-State Circuits Conference (ISSCC), San Francisco, CA, USA, 2016

—, "Understanding Phase Noise in LC VCOs: A Key Problem in RF Integrated Circuits," in IEEE Solid-State Circuits Magazine, vol. 8, no. 4, pp. 81-91, Fall 2016 [https://sci-hub.se/10.1109/MSSC.2016.2573979]

—, Phase Noise in LC Oscillators: From Basic Concepts to Advanced Topologies [https://www.ieeetoronto.ca/wp-content/uploads/2020/06/DL-VCO-short.pdf]

Jun Yin. ISSCC 2025 T10: mm-Wave Oscillator Design

Razavi, Behzad. RF Microelectronics. 2nd ed. Prentice Hall, 2012.

Lacaita, Andrea Leonardo, Salvatore Levantino, and Carlo Samori. Integrated frequency synthesizers for wireless systems. Cambridge University Press, 2007

M. Babaie, M. Shahmohammadi, R. B. Staszewski, (2019) "RF CMOS Oscillators for Modern Wireless Applications" River Publishers [https://www.riverpublishers.com/pdf/ebook/RP_E9788793609488.pdf]

Manetakis, K. (2023). Topics in LC Oscillators: Principles, phase noise, pulling, inductor design. Springer Nature Switzerland Springer. [eetop link]