LC Oscillator

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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

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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]

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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]

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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

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Definitions of Q

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Assuming RLC oscillator waveform is \(V_0\sin\omega_0 t\) and \(\omega_0 = \frac{1}{\sqrt{LC}}\) is resonance frequency. suppose the current through \(R\) is cancelled out by additional \(-R\)

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

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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 \[ Q_3 = \frac{E_{pL} - E_{pC}}{E_d} = \left(\frac{1}{L\omega^2}-C\right)R\omega=\frac{R}{L\omega}\left(1 - \frac{\omega^2}{\omega^2_{SR}}\right) \]

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Makarov, Sergey & Ludwig, Reinhold & Bitar, Joyce. (2016). Practical Electrical Engineering. 10.1007/978-3-319-21173-2. [pdf]

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

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Output Amplitude

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

NMOS Realization

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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

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CMOS Realization

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

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current limited vs voltage limited

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Cross-coupled Differential-pair

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?? Triode MOS noise

Current-biased & voltage-biased DCO

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

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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]

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\[ f=\frac{1}{2\pi\sqrt{L_p C_p}} = \frac{1}{2\pi\sqrt{L_s\frac{Q_L^2+1}{Q_L^2} C_s\frac{Q_C^2}{Q_C^2+1}}} = \frac{1}{2\pi\sqrt{L_sC_s}}\cdot \sqrt{\frac{1+1/Q_c^2}{1+1/Q_L^2}} \] 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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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")))

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Common-Mode Resonance

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]

—, "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]

—, "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]

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

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Tail filter

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]

M. Shahmohammadi, M. Babaie and R. B. Staszewski, "A 1/f Noise Upconversion Reduction Technique for Voltage-Biased RF CMOS Oscillators," in IEEE Journal of Solid-State Circuits, vol. 51, no. 11, pp. 2610-2624, Nov. 2016 [pdf]

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]

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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 [slides,transcript]

—. 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. [pdf]

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

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