Oscillator Phase Noise

Posted on 2024-10-16 Edited on 2026-06-23 In osc

Poddar, Ajay & Rohde, Ulrich & Apte, Anisha. (2013). How Low Can They Go?: Oscillator Phase Noise Model, Theoretical, Experimental Validation, and Phase Noise Measurements. Microwave Magazine, IEEE. [http://time.kinali.ch/rohde/noise/how_low_can_they_go-2013-poddar_rohde_apte.pdf]

F. L. Traversa, M. Bonnin and F. Bonani, "The Complex World of Oscillator Noise: Modern Approaches to Oscillator (Phase and Amplitude) Noise Analysis," in IEEE Microwave Magazine, vol. 22, no. 7, pp. 24-32, July 2021 [https://sci-hub.ru/10.1109/MMM.2021.3069535]

Phase Noise Definition

Eq. (3.25) is widely adopted by industry and academia

using the narrow angle assumption, the two definitions above are equivalent

If the narrow angle condition is not satisfied, however, the two definitions differ

Sam Palermo, ECEN620: Network Theory Broadband Circuit Design Fall 2025, Lecture 7: Voltage-Controlled Oscillators[https://people.engr.tamu.edu/spalermo/ecen620/lecture07_ee620_vcos.pdf]

Phase Noise Profile

Power Spectral Density of Brownian Motion despite non-stationary [https://dsp.stackexchange.com/a/75043/59253]

white noise — $1/f^2$ Phase Noise Profile

Sudhakar Pamarti. CICC 2020 ES2-2: Basics of Closed- and Open-Loop Fractional Frequency Synthesis [https://youtu.be/t1TY-D95CY8]

flicker noise — $1/f^3$ Phase Noise Profile \[ S_{\phi n} = \frac{K}{f}\left(\frac{K_{VCO}}{2\pi f}\right)^2 \propto \frac{1}{f^3} \]

[https://dsp.stackexchange.com/a/75152/59253]

Free-running Oscillator

Note that $f_{min}$ is related to the observation time. The longer we observe the device under test, the smaller $f_{min}$ must be

Ali Sheikholeslami ISSCC 2008 T5: Basics of Chip-to-Chip and Backplane Signaling [https://www.nishanchettri.com/isscc-slides/2008%20ISSCC/Tutorials/T10_Pres.pdf]

B. Casper and F. O'Mahony, "Clocking Analysis, Implementation and Measurement Techniques for High-Speed Data Links-A Tutorial," in IEEE Transactions on Circuits and Systems I. [https://people.engr.tamu.edu/spalermo/ecen689/clocking_analysis_hs_links_casper_tcas1_2009.pdf]

Leeson's Model — LTI

M.H. Perrott, Short Course On Phase-Locked Loops and Their Applications Day 2, AM Lecture Basic Building Blocks Voltage-Controlled Oscillators [https://www.cppsim.com/PLL_Lectures/day2_am.pdf]

—, 6.976 High Speed Communication Circuits and Systems Lecture 12 Noise in Voltage Controlled Oscillators [https://ocw.mit.edu/courses/6-976-high-speed-communication-circuits-and-systems-spring-2003/ceb3d539691d5393a29af71ae98afb62_lec12.pdf]

Leeson's model is outcome of linearized VCO noise analysis

Assuming voltage noise tone $(\omega_0+\omega_m)$ and $(\omega_0-\omega_m)$ are independent and symmetric

two_offset_folding_into_pm_sidebands

Leeson's limitations

Hajimiri's ISF— LTV in Time Domain

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 [paper], [slides]

—, RFIC 2024 Technical Lecture: Noise in Oscillators from Understanding to Design

Thomas H. Lee. Linearity, Time-Variation, Phase Modulation and Oscillator Phase Noise [https://class.ece.iastate.edu/djchen/ee507/PhaseNoiseTutorialLee.pdf]

Aditya Varma Muppala, [https://adityamuppala.github.io/assets/Notes_YouTube/Oscillators_ISF_model.pdf]

Pure sinusoidal voltage

Consider the ideal parallel LC network — pure sinusoid wave

Suppose a current pulse with area $q$ suddenly changes the charge across the capacitor, its voltage changes by $\Delta v_c=\Delta q/C$

phase_kick_decomposition_radial_tangential

Decompose the horizontal kick $r=(x,0) $ and $\Delta x=\Delta v_C/A_0$, with tangential direction $\hat t=(-\sin\theta,\cos\theta)$ and radial direction $\hat r=(\cos\theta,\sin\theta)$ \[ \Delta \phi = \arctan\left(\frac{\Delta\vec r\cdot \hat t}{1 + \Delta\vec r\cdot \hat r}\right) = \arctan\left(\frac{-\Delta x \sin \theta}{1 + \Delta x \cos \theta}\right)\approx -\frac{\Delta v_C}{A_0} \sin(\omega_0 \tau) \]

Therefore, the ISF of an ideal parallel LC resonator can be expressed as $\boxed{\Gamma(\omega \tau)=-\sin(\omega_0 \tau)}$, which is independent of peak voltage value $A_0$

White-noise Folding

Suppose a low frequency sinusoidal perturbation current $i(t) = I_m \cos[(m\omega_0 +\Delta \omega)t]$,

\[\begin{align} \phi(t) &= \frac{1}{q_\text{max}}\left[\frac{C_0}{2}\int_{-\infty}^t I_m\cos((m\omega_0 +\Delta \omega)\tau)d\tau + \sum_{n=1}^\infty C_n\int_{-\infty}^t I_m\cos((m\omega_0 +\Delta \omega)\tau)\cos(n\omega_0\tau)d\tau\right] \\ &= \frac{I_m}{q_\text{max}}\left[\frac{C_0}{2}\int_{-\infty}^t \cos((m\omega_0 +\Delta \omega)\tau)d\tau + \sum_{n=1}^\infty C_n\int_{-\infty}^t \frac{\cos((m\omega_0 + \Delta \omega+ n\omega_0)\tau)+ \cos((m\omega_0+\Delta \omega - n\omega_0)\tau)}{2}d\tau\right] \end{align}\]

If $m=0$ \[ \phi(t) \approx \frac{I_0C_0}{2q_\text{max}\Delta \omega}\sin(\Delta\omega t) \] If $m\neq 0$ and $m=n$ \[ \phi(t) \approx \frac{I_mC_m}{2q_\text{max}\Delta \omega}\sin(\Delta\omega t) \]

When performing the phase noise computation integral, there will be a negligible contribution from all terms, other than $n=m$

apply equation (18) derived from sinusoidal to white noise

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]

L. Lu, Z. Tang, P. Andreani, A. Mazzanti and A. Hajimiri, "Comments on “Comments on “A General Theory of Phase Noise in Electrical Oscillators””," 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]

Noise power around the frequency $\color{blue}n\omega_0 + \Delta\omega$ causes two equal sidebands at $\omega_0 \pm \Delta\omega$. However, the noise power at $\color{blue}n\omega_0 - \Delta\omega$ has a similar effect as mentioned in the paper. Therefore, twice the power of noise at $n\omega_0 + \Delta\omega$ should be taken into account

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

	$m = 0$	$m\gt 0$ & $m\omega_0+\Delta \omega$	$m\gt 0$ & $m\omega_0-\Delta \omega$
$\phi(t)$	$\frac{I_0C_0}{2q_\text{max}\Delta \omega}\sin(\Delta\omega t)$	$\frac{I_mC_m}{2q_\text{max}\Delta \omega}\sin(\Delta\omega t)$	$\frac{I_mC_m}{2q_\text{max}\Delta \omega}\sin(\Delta\omega t)$
$P_{SBC}(\Delta \omega)$	$10\log(\frac{I_0^2C_0^2}{16q_\text{max}^2\Delta \omega^2})$	$10\log(\frac{I_m^2C_m^2}{16q_\text{max}^2\Delta \omega^2})$	$10\log(\frac{I_m^2C_m^2}{16q_\text{max}^2\Delta \omega^2})$

\[\begin{align} \mathcal{L}\{\Delta \omega\} &= 10\log\left(\frac{I_0^2C_0^2}{16q_\text{max}^2\Delta \omega^2} + 2\frac{I_m^2C_m^2}{16q_\text{max}^2\Delta \omega^2}\right) = 10\log\left(\frac{\overline{i_n^2/\Delta f}\cdot \frac{C_0^2}{2} }{4q_\text{max}^2\Delta \omega^2} + \frac{\overline{i_n^2/\Delta f}\cdot\sum_{m=1}^\infty C_m^2 }{4q_\text{max}^2\Delta \omega^2}\right) \\ &= 10\log \frac{\overline{i_n^2/\Delta f}(C_0^2/2+\sum_{m=1}^\infty C_m^2)}{4q_\text{max}^2\Delta \omega^2} = 10\log \frac{\overline{i_n^2/\Delta f}\cdot \Gamma_\text{rms}^2}{2q_\text{max}^2\Delta \omega^2} \end{align}\]

[pdf]

$1/f$-noise Upconversion

Suppose $c_0\neq 0$, corresponding phase noise in response to injected noise $i_n(t)$ is equal to:

\[ \phi_{n,c_0} = \int_{-\infty}^t c_0 i_n(\tau) d\tau \qquad \boxed{S_{\phi n,c_0}(f) = \frac{c_0^2}{\omega^2}S_i(f)= \frac{\mathcal{\Gamma}_\text{dc}^2}{\omega^2}S_i(f)} \]

Cyclostationary Noise Sources

Cyclostationary noise can be viewed as stationary noise, $i_{n0}(t)$, multiplied by a periodic envelope, $\alpha(\omega_0 t)$.

Effective ISF — ISF multiplied with Noise Modulating Function (NMF)

Diff. Pair Noise with ISF

Given $\Gamma_{MOS}$ shown as above slide \[ F_{rms,MOS}^2 = \frac{1/4\cdot T_\text{w}}{T_0/2} = \frac{T_\text{w}}{2T_0} \]

Murphy's Model — LTV in Frequency Domain

C. Samori, A. L. Lacaita, F. Villa and F. Zappa, "Spectrum folding and phase noise in LC tuned oscillators," in IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, vol. 45, no. 7, pp. 781-790, July 1998 [https://sci-hub.ru/10.1109/82.700925]

D. Murphy, J. J. Rael and A. A. Abidi, "Phase Noise in LC Oscillators: A Phasor-Based Analysis of a General Result and of Loaded Q ," in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 57, no. 6, pp. 1187-1203, June 2010 [https://sci-hub.ru/10.1109/TCSI.2009.2030110]

The differential pair plays two distinct roles. Toward its own noise it acts as a sampling gate — each transistor contributes only within the short conduction windows at the zero crossings — and the resulting injection is almost purely phase-modulating.

Toward the tail noise it acts as a single-balanced mixer, commutating the upstream current with a square wave and folding it as a single sideband that splits equally into AM and PM

Diff. Pair Noise

For diff. pair noise, the diff. pair is its own noise gate

A reference pulse train centered at $t=0$ would have plain positive coefficients ($\tfrac{T_w}{T_0}$, with $\operatorname{sinc} \approx 1$). \[ c_n = \frac{1}{2}\cdot \frac{T_w}{T_0/2}\operatorname{sinc}\left(\frac{nT_w}{T_0/2}\right)=\frac{T_w}{T_0}\operatorname{sinc}\left(\frac{2nT_w}{T_0}\right) \] The real injection sits at $t=T_0/4$, and since the pulse-train period is $P=T_0/2$, that offset is exactly $P/2$ — a half-period shift. Every coefficient therefore picks up $(-1)^m$ — $e^{j2m\omega_0\cdot T_0/4}=e^{jm\pi}$ \[ c_{2m} = \underbrace{(-1)^m}_{\text{position}}\, \underbrace{\frac{T_w}{T_0}}_{\text{area / period}}\, \underbrace{\operatorname{sinc}\!\left(\frac{2mT_w}{T_0}\right)}_{\to\,1 \text{ as } T_w \ll T_0} \]

Consequently, excess noise arises solely from the components at $\omega_0\pm \omega_m$, since all other spectral components lie outside the tank bandwidth and are therefore suppressed by the resonator's frequency-selective filtering

With Cyclostationary Noise (Modulated Noise) [https://raytroop.github.io/2024/04/27/noise/#cyclostationary-noise-modulated-noise]

For one MOS, Two-Sided PSD is \[ S_O = S_I \cdot \mathcal{D}\cdot \mathcal{h}^2 = 2kT\gamma g_m\cdot \frac{2T_W}{T_0}\cdot\frac{1}{4} = \frac{T_W}{2T_0}\cdot 2kT\gamma g_m \] yield One-Sided PSD of one MOS \[ S_O' = \textcolor{blue}{\frac{T_W}{2T_0}}\cdot 4kT\gamma g_m \]

The single-tone analysis establishes that the differential-pair current is injected as almost pure phase noise, while summing the white-noise power over all harmonics of the gating function ($=T_W/2T_0 $) sets its magnitude; together these yield the differential-pair contribution to the oscillator phase noise.

phase noise is independent of the transconductance of the transistors

Tail Noise

For the tail noise, the diff. pair is a mixer

\[ V_{AM} = \tfrac{1}{2}\big(C_+ + \overline{C}_-\big) = \tfrac{1}{2}\Big(\tfrac{c_1^* i}{2} + \tfrac{c_3^* i}{2}\Big)\qquad V_{PM} = \tfrac{1}{2}\big(C_+ - \overline{C}_-\big) = \tfrac{1}{2}\Big(\tfrac{c_1^* i}{2} - \tfrac{c_3^* i}{2}\Big) \] $V_{AM}\approx V_{PM}$ for a square wave $|c_1|=3|c_3|$ — modulated tail noise is divided into AM and phase noise almost equally \[ S_{I,PN} = S_{nI,T} \cdot \mathcal{D}\cdot \mathcal{h}^2 \cdot \frac{1}{2} = S_{nI,T} \cdot 1 \cdot \frac{1}{4}\cdot \frac{1}{2} = \boxed{\frac{1}{8}\cdot S_{nI,T}} \] The commutation folds tail noise as a (near) single sideband, which is equivalent to equal AM and PM — and only the PM half counts toward phase noise

noise at $2\omega_0 \pm\omega_m$ dominate phase noise due to $|c_1|, |c_3| \gg |c_{2m+1}| \space\space\space\space \forall m>1$

Demir's Model — NLTV

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]

Demir's theory is essentially Floquet theory applied to the limit cycle of an autonomous oscillator, and the PPV is one specific Floquet vector

PPV (Perturbation Projection Vector)

A. Demir and J. Roychowdhury, "A reliable and efficient procedure for oscillator PPV computation, with phase noise macromodeling applications," in IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, vol. 22, no. 2, pp. 188-197, Feb. 2003 [https://sci-hub.se/10.1109/TCAD.2002.806599]

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 [youtu.be, notes]

Limit Cycles

[https://adityamuppala.github.io/assets/Notes_YouTube/MMIC_Limit_Cycles.pdf]

Nonlinear Dynamics

Lorentzian spectrum

We typically use the two spectra, $S_{\phi n}(f)$ and $S_{out}(f)$, interchangeably, but we must resolve these inconsistencies. voltage spectrum is called Lorentzian spectrum

The periodic signal $x(t)$ can be expanded in Fourier series as:

Assume that the signal is subject to excess phase noise, which is modeled by adding a time-dependent noise component $\alpha(t)$. The noisy signal can be written $x(t+\alpha(t))$, the added excess phase $\phi(t)= \frac{\alpha(t)}{\omega_0}$

The autocorrelation of the noisy signal is by definition:

The autocorrelation averaged over time results in:

By taking the Fourier transform of the autocorrelation, the spectrum of the signal $x(t + \alpha(t))$ can be expressed as

It is also interesting to note how the integral in Equation 9.80 around each harmonic is equal to the power of the harmonic itself $|X_n|^2$

The integral $S_x(f)$ around harmonic is \[\begin{align} P_{x,n} &= \int_{f=-\infty}^{\infty} |X_n|^2\frac{\omega_0^2n^2c}{\frac{1}{4}\omega_0^4n^4c^2+(\omega +n\omega_0)^2}df \\ &= |X_n|^2\int_{\Delta f=-\infty}^{\infty}\frac{2\beta}{\beta^2+(2\pi\cdot\Delta f)^2}d\Delta f \\ &= |X_n|^2\frac{1}{\pi}\arctan(\frac{2\pi \Delta f}{\beta})|_{-\infty}^{\infty} \\ &= |X_n|^2 \end{align}\]

The phase noise does not affect the total power in the signal, it only affects its distribution

Without phase noise, $S_v(f)$ is a series of impulse functions at the harmonics of $f_o$.
With phase noise, the impulse functions spread, becoming fatter and shorter but retaining the same total power

[https://community.cadence.com/cadence_technology_forums/f/rf-design/51484/comparing-transient-noise-pnoise-and-pnoise-with-lorentian-approximation-of-a-ring-oscillator/1382911]

Phase Perturbation

Chembiyan T, "Brownian Motion And The Oscillator Phase Noise" [link]

—, "Jitter and Phase Noise in Oscillators" [link]

—, "Jitter and Phase Noise in Phase Locked Loops" [link]

—, "PLLs and reference spurs" [link]

w/ stationary noise with Gaussian PDF

If keep $\phi_{rms}$ in $R_x(\tau)$, i.e. \[ R_x(\tau)=\frac{A^2}{2}e^{-\phi_{rms}^2}\cos(2\pi f_0 \tau)e^{R_\phi(\tau)}\approx \frac{A^2}{2}e^{-\phi_{rms}^2}\cos(2\pi f_0 \tau)(1+R_\phi(\tau)) \] The PSD of the signal is \[ S_x(f) = \mathcal{F} \{ R_x(\tau) \} = \frac{P_c}{2}e^{-\phi_{rms}^2}\left[S_\phi(f+f_0)+S_\phi(f-f_0)\right] + \frac{P_c}{2}e^{-\phi_{rms}^2}\left[\delta(f+f_0)+\delta(f-f_0)\right] \] ❗❗above Eq isn't consistent with stationary white noise process - the following section

w/ stationary white noise

Assuming that the delay line is noiseless

Expanding the cosine function we get \[\begin{align} R_y(t,\tau) &= \frac{A^2}{2}\left\{\cos(2\pi f_0\tau)E[\cos(\phi(t)-\phi(t-\tau))] - \sin(2\pi f_0\tau)E[\sin(\phi(t)-\phi(t-\tau))]\right\} \\ &+ \frac{A^2}{2}\left\{\cos(4\pi f_0(t+\tau/2-T_D))E[\cos(\phi(t)+\phi(t-\tau))] - \sin(4\pi f_0(t+\tau/2-T_D))E[\sin(\phi(t)+\phi(t-\tau))] \right\} \end{align}\]

where, both the process $\phi(t)-\phi(t-\tau)$ and $\phi(t)+\phi(t-\tau)$ are independent of time $t$, i.e. $E[\cos(\phi(t)+\phi(t-\tau))] = m_{\cos+}(\tau)$, $E[\cos(\phi(t)-\phi(t-\tau))] = m_{\cos-}(\tau)$, $E[\sin(\phi(t)+\phi(t-\tau))] = m_{\sin+}(\tau)$ and $E[\sin(\phi(t)-\phi(t-\tau))] = m_{\sin-}(\tau)$

we obtain \[\begin{align} R_y(t,\tau) &= \frac{A^2}{2}\left\{\cos(2\pi f_0\tau)m_{\cos-}(\tau) - \sin(2\pi f_0\tau)m_{\sin-}(\tau)\right\} \\ &+ \frac{A^2}{2}\left\{\cos(4\pi f_0(t+\tau/2-T_D))m_{\cos+}(\tau) - \sin(4\pi f_0(t+\tau/2-T_D))m_{\sin+}(\tau) \right\} \end{align}\]

The second term in the above expression is periodic in $t$ and to estimate its PSD, we compute the time-averaged autocorrelation function \[ R_y(\tau) = \frac{A^2}{2}\left\{\cos(2\pi f_0\tau)m_{\cos-}(\tau) - \sin(2\pi f_0\tau)m_{\sin-}(\tau)\right\} \]

After nontrivial derivation

w/ Weiner process

The phase process $\phi(t)$ is also gaussian but with an increasing variance which grows linearly with time $t$

\[\begin{align} R_y(t,\tau) &= \frac{A^2}{2}\left\{\cos(2\pi f_0\tau)E[\cos(\phi(t)-\phi(t-\tau))] - \sin(2\pi f_0\tau)E[\sin(\phi(t)-\phi(t-\tau))]\right\} \\ &+ \frac{A^2}{2}\left\{\cos(4\pi f_0(t+\tau/2-T_D)E[\cos(\phi(t)+\phi(t-\tau))] - \sin(4\pi f_0(t+\tau/2-T_D)E[\sin(\phi(t)+\phi(t-\tau))] \right\} \end{align}\]

The spectrum of $y(t)$ is determined by the asymptotic behavior of $R_y(t,\tau)$ as $t\to \infty$

❗❗ $\lim_{t\to\infty}R_y(t,\tau)$ rather than time-averaged autocorrelation function of cyclostationary process, ref. Demir's paper

We define $\zeta(t, \tau)=\phi(t)+\phi(t-\tau) = \phi(t)-\phi(t-\tau) + 2\phi(t-\tau)$, the expected value of $\zeta(t,\tau)$ is 0, the variance is $\sigma_{\zeta}^2=(k\sigma)^2(\tau + 4(t-\tau))=(k\sigma)^2(4t-3\tau)$ \[ E[\cos(\zeta(t,\tau))]=\frac{1}{\sqrt{2\pi \sigma_{\zeta}^2}}\int_{-\infty}^{\infty}e^{-\zeta^2/2\sigma_{\zeta}^2}\cos(\zeta)d\zeta = e^{-\sigma_{\zeta}^2/2}=e^{-(k\sigma)^2(4t-\tau)} \] i.e., $\lim _{t\to \infty} E[\cos(\zeta(t,\tau))] = \lim_{t\to \infty}e^{-(k\sigma)^2(4t-\tau)} = 0$

For $E[\sin(\zeta(t,\tau))]$, we have \[ E[\sin(\zeta(t,\tau))] = \frac{1}{\sqrt{2\pi \sigma_{\zeta}^2}}\int_{-\infty}^{\infty}e^{-\zeta^2/2\sigma_{\zeta}^2}\sin(\zeta)d\zeta \] i.e., $E[\sin(\zeta(t,\tau))]$ is odd function, therefore $E[\sin(\zeta(t,\tau))]=0$

Finally, we obtain

Common-Mode Resonance

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]

A. Bevilacqua and P. Andreani, "An Analysis of 1/f Noise to Phase Noise Conversion in CMOS Harmonic Oscillators," in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 59, no. 5, pp. 938-945, May 2012

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]

—, "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]

M. Shahmohammadi, M. Babaie and R. B. Staszewski, "25.4 A 1/f noise upconversion reduction technique applied to Class-D and Class-F oscillators," 2015 IEEE International Solid-State Circuits Conference - (ISSCC) Digest of Technical Papers, San Francisco, CA, USA, 2015 [https://sci-hub.ru/10.1109/ISSCC.2015.7063117]

—, "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 [https://pure.tudelft.nl/ws/portalfiles/portal/30880387/07571191.pdf]

Yunbo Huang, Zunsong Yang*, et al., "A 7.0-to-8.6GHz Balanced Class-F-1 VCO with a Trifilar Transformer-Based Tank Achieving 194.5dBc/Hz FoM," IEEE MTT-S Radio Frequency Integrated Circuits (RFIC), June 2026

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 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

ISF & PPV simulation

David Dolt. ECEN 620 Network Theory - Broadband Circuit Design: "VCO ISF Simulation" [https://people.engr.tamu.edu/spalermo/ecen620/ISF_SIM.pdf]

Hu, Yizhe. (2019). A Simulation Technique of Impulse Sensitivity Function (ISF) Based on Periodic Transfer Function (PXF). 10.13140/RG.2.2.32151.60323. [link]

Aditya Varma Muppala, ISF Simulation in Cadence Using Transient Analysis | Oscillators 07 | MMIC 12 [https://youtu.be/yiMn2rCtTXY]

Aditya Varma Muppala, Fast Simulation of ISF and PPV using PSS and PXF in Cadence | Oscillators 12 | MMIC 19 [https://youtu.be/Lu6VEWEEdxo]

S. Galeone and M. P. Kennedy, "A comparison of simulation strategies for estimating phase noise in oscillators," 2017 13th Conference on Ph.D. Research in Microelectronics and Electronics (PRIME), Giardini Naxos - Taormina, Italy, 2017

PPV values from pss/pnoise simulation in spectreRF [https://community.cadence.com/cadence_technology_forums/f/rf-design/35062/ppv-values-from-pss-pnoise-simulation-in-spectrerf]

ISF Function Extraction in Cadence Virtuoso [https://community.cadence.com/cadence_technology_forums/f/custom-ic-design/43969/isf-function-extraction-in-cadence-virtuoso]

ISF using Transient Analysis

To compare the ring oscillator and VCO the total injected charge to both should be the same

ISF using PSS + PXF

TODO 📅

PPV using PSS + PXF

TODO 📅

References

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

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, ISSCC2016 T1 "Tutorial: Understanding Phase Noise in LC VCOs"

—, "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.ru/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]

Akihide Sai, Toshiba. ISSCC 2023 T5: All-digital PLLs From Fundamental Concepts to Future Trends

A. Hajimiri, RFIC2024 "Noise in Oscillators from Understanding to Design"

Antonio Liscidini, ESSCIRC 2019 Tutorials: Phase Noise in Wireless Applications [https://youtu.be/nGmQ0JdoSE4]

Aditya Varma Muppala. Oscillators [https://youtube.com/playlist?list=PL9Trid0A4Da2fOmYTEjhAnUkGPxyiH7H6&si]

P.E. Allen - 2003. ECE 6440 - Frequency Synthesizers: Lecture 160 – Phase Noise - II [https://pallen.ece.gatech.edu/Academic/ECE_6440/Summer_2003/L160-PhNoII(2UP).pdf]

ISF, PPV, and Flicker-Noise Analysis

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]

—, "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]

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]

Bae, Woorham, and Deog-Kyoon Jeong. Analysis and Design of CMOS Clocking Circuits for Low Phase Noise. Institution of Engineering and Technology, 2020.

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

Hegazi, Emad, Asad Abidi, and Jacob Rael. The Designer's Guide to High-purity Oscillators. [New York]: Kluwer Academic Publishers, 2005. The Designer's Guide to High-Purity Oscillators [pdf]

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]

Jaeha Kim. Lecture 8. Special Topics: Design Trade -Offs in LC -Tuned Oscillators [https://ocw.snu.ac.kr/sites/default/files/NOTE/7033.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]

Mathematical Background

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]

Cadence Blog, "Resonant Frequency vs. Natural Frequency in Oscillator Circuits" [link]

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]

	\(m = 0\)	\(m\gt 0\) & \(m\omega_0+\Delta \omega\)	\(m\gt 0\) & \(m\omega_0-\Delta \omega\)
\(\phi(t)\)	\(\frac{I_0C_0}{2q_\text{max}\Delta \omega}\sin(\Delta\omega t)\)	\(\frac{I_mC_m}{2q_\text{max}\Delta \omega}\sin(\Delta\omega t)\)	\(\frac{I_mC_m}{2q_\text{max}\Delta \omega}\sin(\Delta\omega t)\)
\(P_{SBC}(\Delta \omega)\)	\(10\log(\frac{I_0^2C_0^2}{16q_\text{max}^2\Delta \omega^2})\)	\(10\log(\frac{I_m^2C_m^2}{16q_\text{max}^2\Delta \omega^2})\)	\(10\log(\frac{I_m^2C_m^2}{16q_\text{max}^2\Delta \omega^2})\)