β: This English translation is in beta — the Traditional-Chinese original is the authoritative version.

Jitter and Phase Noise in Ring Oscillators

Prerequisites (recommended reading order): first digest paper_001 (ISF, $\Gamma_{rms}^2/q_{max}^2$, the symmetry rule) — every conclusion on this page is [P1]'s ISF applied to the ring. For the time-/frequency-domain language of jitter see psd_phase_noise_jitter.

[P2] applies the ISF framework of [P1] to the ring oscillator. It answers three very practical questions: (1) how does the long-term jitter of a free-running ring grow with time? (2) what effect does the number of stages $N$ have on phase noise? (3) why does waveform symmetry suppress close-in noise? The answers are, respectively, $\sigma_{\Delta t}=\kappa\sqrt{\Delta t}$, $\Gamma_{rms}\propto N^{-3/2}$ (with "nearly $N$-independent at fixed power and frequency"), and the symmetry experiment of Fig. 17.

Citation

[P2] A. Hajimiri, S. Limotyrakis, and T. H. Lee, "Jitter and Phase Noise in Ring Oscillators," IEEE J. Solid-State Circuits, vol. 34, no. 6, pp. 790–804, Jun. 1999. (file jitter_ring.pdf, paper_002)

One-sentence contribution

Applying [P1]'s ISF to the ring oscillator yields closed forms for jitter and phase noise, the $\Gamma_{rms}\propto N^{-3/2}$ scaling, and the counter-intuitive conclusion that "at fixed $f_0$ and power, the phase noise/jitter of a single-ended ring is nearly independent of the number of stages $N$" (claim C7, C8).

Why this paper matters

An LC oscillator needs an inductor — large area, hard to integrate; a ring oscillator is all inverters: small area, easy to integrate, wide tuning range — the most common VCO in PLLs/CDRs. But a ring's phase noise is usually much worse than an LC's — [P2] uses the ISF to explain why, and gives actionable design rules:

• It ties the ring's jitter to the same $\Gamma_{rms}^2/q_{max}^2$ ratio as phase noise (claim C6), unifying "time-domain jitter" and "frequency-domain phase noise" under the ISF framework.
• It settles a commonly misunderstood question: "does adding more stages $N$ to a ring make it better?" Under the constraint of fixed power and frequency the answer is "almost no difference" (claim C7) — more stages shrink $\Gamma_{rms}$, but each stage's swing shrinks and there are more devices; the effects cancel.
• It confirms [P1]'s symmetry rule with measurements (Fig. 17): tuning the control voltage to the point of symmetric rise/fall produces a minimum in phase noise (claim C4).

Main assumptions

Per paper_metadata (paper_002.assumptions):

1. The same LTV/ISF small-perturbation assumptions as [P1].
2. Per-stage device noise is white (plus a 1/f component handled via symmetry).
3. Identical stages; delay and noise add independently at every transition.

Physical intuition: nearly all of a ring's energy is injected in the instant of a transition (edge flip), so its ISF is not the smooth $-\sin$ of an LC but a set of sharp peaks concentrated at the transitions ([P2] Fig. 5). The sensitive spots — where a kick hurts phase the most — sit on those peaks. The more stages, the smaller the fraction of the full period a single transition occupies, and the smaller the rms ISF.

Key equations

Eq.(8): accumulated jitter (the random-walk fingerprint)

Original formula ([P2] Eq.(8), p.792; κ from Eq.(12), p.793):

$$\sigma_{\Delta t}=\kappa\sqrt{\Delta t}$$

Meaning: for two edges of a free-running oscillator separated by $\Delta t$, the standard deviation of the timing error is proportional to $\sqrt{\Delta t}$ — the random-walk fingerprint of an oscillator with "no absolute time reference" (claim C6). $\kappa$ is a device-dependent proportionality constant with units of $\sqrt{\text{s}}$.

Step-by-step derivation: each transition injects an independent, zero-mean timing perturbation with variance $\sigma_{step}^2$. Over $\Delta t$ there are about $M=\Delta t/T$ transitions; independent quantities add in variance:

$$\begin{aligned} \sigma_{\Delta t}^2 &= M\,\sigma_{step}^2 = \frac{\Delta t}{T}\,\sigma_{step}^2 \\ \Rightarrow\quad \sigma_{\Delta t} &= \underbrace{\frac{\sigma_{step}}{\sqrt{T}}}_{\equiv\,\kappa}\sqrt{\Delta t}=\kappa\sqrt{\Delta t}. \end{aligned}$$

Dimension check: $\kappa$ is $\sqrt{\text{s}}$ and $\sqrt{\Delta t}$ is $\sqrt{\text{s}}$; their product is $\text{s}$ ✓. Cross-check against [P1] in the frequency domain: $\sigma_{\Delta t}\propto\sqrt{\Delta t}$ corresponds to 1/f² phase noise (the two are the time-/frequency-domain faces of the same thing).

Numerical example: the toy ring sets per-edge $\sigma_{step}=50$ fs (see the parameters of ring_oscillator_timing_noise_accumulation.png). Accumulated jitter over $\Delta t=1$ µs (about 5000 periods at $f_0=5$ GHz): first find $\kappa$. With $T=200$ ps, $\kappa=50\text{fs}/\sqrt{200\text{ps}}=50\times10^{-15}/\sqrt{2\times10^{-10}}=3.54\times10^{-9}\ \sqrt{\text{s}}$, so $\sigma_{\Delta t}=3.54\times10^{-9}\times\sqrt{10^{-6}}=3.54\ \text{ps}$. Intuition: the longer the separation, the larger the drift — but it grows only slowly, as $\sqrt{\Delta t}$.

Python verification:

import numpy as np
from simulations.common.oscillator_models import accumulated_jitter_curve

# toy random walk: each edge adds an independent 50 fs timing perturbation
lags, sigma = accumulated_jitter_curve(f0=5e9, sigma_edge=50e-15, max_lag_periods=500, n_trials=2000)
# expect sigma(lag) ~ sigma_edge * sqrt(lag) -> log-log slope 0.5
slope = np.polyfit(np.log(lags[1:]), np.log(sigma[1:]), 1)[0]
print(round(slope, 2))  # -> 0.50

The full toy derivation is in lab_03_ring_oscillator_toy_model (pedagogical toy model, not transistor-level).

Eq.(11)–(12): the jitter constant κ and its relation to the ISF (verified ✓)

Original formula ([P2] Eq.(11)–(12), p.793, proportionality):

$$\kappa^2\;\propto\;\frac{\Gamma_{rms}^2}{q_{max}^2}\cdot\frac{\overline{i_n^2}}{\Delta f}$$

Meaning: the jitter proportionality constant $\kappa$ is set by exactly the same $\Gamma_{rms}^2/q_{max}^2$ ratio as phase noise (claim C6). This ties time-domain jitter and frequency-domain phase noise to the same ISF quantity: the knobs that lower phase noise lower jitter as well.

Verified: [P2] Eq.(12), p.793 gives $\kappa=\frac{\Gamma_{rms}}{q_{max}}\sqrt{\tfrac12\tfrac{\overline{i_n^2}}{\Delta f}}$ (verified verbatim against the original PDF rendering); it shares the same $\Gamma_{rms}^2/q_{max}^2$ ratio with phase noise (claim C6).

Eq.(14): ring frequency vs number of stages

Original formula ([P2] Eq.(14), p.794):

$$f_0=\frac{1}{2N\tau_D}$$

Meaning: an $N$-stage ring with per-stage delay $\tau_D$ oscillates at this frequency. The factor of 2 comes from the signal having to travel around the ring twice per period (one lap inverts; a second lap returns it in phase) to complete one full cycle.

Dimension check: $1/(N\cdot\text{s})=\text{Hz}$ ✓ ($N$ dimensionless).

Numerical example: for a 5-stage ring at $f_0=5$ GHz, the per-stage delay is $\tau_D=1/(2\times5\times5\times10^9)=2\times10^{-11}\ \text{s}=20$ ps. Doubling to $N=10$ while keeping 5 GHz halves the per-stage delay to 10 ps — this is exactly where "at fixed frequency, larger $N$ forces faster, smaller-swing stages" comes from, leading into the N-independence below.

Eq.(16): rms-ISF scaling with the number of stages (re-verified in v7 ✓)

Original formula ([P2] Eq.(16), p.794; the radical covers only the constant term):

$$\Gamma_{rms}=\sqrt{\dfrac{2\pi^2}{3\eta^3}}\;\dfrac{1}{N^{1.5}}$$

where $\eta$ is the frequency proportionality constant (Eq.(14)–(15): $\hat t_D=\eta/f_{max}$, $2\pi=2N\eta/f_{max}$); at $\eta=0.75$, $\sqrt{2\pi^2/(3\times0.75^3)}\approx3.95\approx4$, i.e. $\Gamma_{rms}\approx4/N^{1.5}$, which is the solid line in [P2] Fig. 8 — the radical covers only the constant, and $1/N^{1.5}$ sits outside it.

Meaning: $\Gamma_{rms}\propto N^{-3/2}$ (i.e. $\Gamma_{rms}^2\propto N^{-3}$). Intuition: with more stages, each transition occupies a narrower "sensitive window" of the $2\pi$ period and the peaks get shorter, so the rms naturally shrinks.

[P2] Eq.(16), p.794 (re-verified in v7: the radical covers only the constant, $\Gamma_{rms}\propto N^{-3/2}$; triple-confirmed by the prose's $4/N^{1.5}$@$\eta=0.75$ and App.B Eq.(55). v3 had misread this as $N^{-3/4}$): three independent lines of evidence — (1) the paper's own text (p.794, the paragraph after Eq.16) states "the $1/N^{1.5}$ dependence of $\Gamma_{rms}$"; (2) the $\eta=0.75$ numerical anchor: the text says "solid line = $\Gamma_{rms}\approx4/N^{1.5}$, obtained from (16) for $\eta=0.75$", and $\sqrt{2\pi^2/(3\times0.75^3)}=3.95\approx4$ ✓ (if $N^{1.5}$ were inside the radical this would give $4/N^{0.75}$, contradicting the text); (3) independent algebra in App.B Eq.(52)+(54) (p.803): $\Gamma_{rms}^2=(1/3\pi)(1/f'_{rise})^3(1+A^3)$, $2\pi=\eta N(1+A)/f'_{rise}$, which combine to give $\Gamma_{rms}^2=(2\pi^2/3\eta^3)\cdot[4(1+A^3)/(1+A)^3]\cdot N^{-3}$; at $A=1$ the bracket equals 1, so $\Gamma_{rms}^2\propto N^{-3}\Rightarrow\Gamma_{rms}\propto N^{-3/2}$ ✓. All three point to $N^{-3/2}$; there is no real "formula-vs-text" inconsistency — it was a prior misreading of the radical's scope.

Eq.(23): ring white-noise phase-noise FOM and N-independence (prefactor corrected to 8/(3η) and verified)

Original formula ([P2] Eq.(23), p.796; the $V_T=0$ lower bound is Eq.(25)):

$$\mathcal{L}\{\Delta f\}=\frac{8}{3\eta}\cdot\frac{kT}{P}\cdot\frac{V_{DD}}{V_{char}}\cdot\left(\frac{f_0}{\Delta f}\right)^2 \qquad\Big(\min_{V_T=0}:\ \frac{16\gamma}{3\eta}\cdot\frac{kT}{P}\cdot\frac{f_0^2}{\Delta f^2}\Big)$$

where $\gamma$ is the MOSFET channel thermal-noise coefficient ($2/3$ long-channel, larger for short-channel), $V_{char}$ is the device's characteristic voltage (long-channel $\approx\Delta V/\gamma$), $P$ is the power dissipation (Eq.(21): $P=2\eta N V_{DD}q_{max}f_0$), and the per-stage noise is given by Eq.(17),(18) $\overline{i_n^2}/\Delta f=4kT\gamma\mu C_{ox}(W/L)\Delta V$.

Meaning: ring white-noise phase noise collapses into a figure of merit — only $kT/P$, the voltage ratio $V_{DD}/V_{char}$, and $(f_0/\Delta f)^2$ appear. Key conclusion (claim C7): $N$ is entirely absent from Eq.(23) — at fixed $f_0$ and power $P$, the phase noise of a single-ended ring is independent of the number of stages $N$.

Why N-independent: microscopically, raising $N$ lowers $\Gamma_{rms}$ (Eq.16) but simultaneously lowers each stage's swing $q_{max}$ and adds more noisy stages; [P2] shows these effects cancel exactly at fixed $P$, $f_0$, so Eq.(23) contains no $N$. Hence "how many stages should my ring have" is not decided by phase noise, but by phase margin, tuning range, area, quadrature needs, and other considerations.

Correction note (v3): the prefactor of [P2] Eq.(23) is $8/(3\eta)$ ($\eta$ being the stage-delay proportionality constant of Eq.14, $\approx1$); $\gamma$ enters only through $V_{char}=\Delta V/\gamma$. (v2 mistakenly changed it to $8/(3\gamma)$ and mislabeled it "verified verbatim"; v3 corrected it against the original PDF p.796.) The $V_T=0$ lower bound is accordingly corrected to $16\gamma/(3\eta)$. $\gamma$ (noise coefficient) and $\eta$ (frequency proportionality constant, Eq.14) are different quantities — do not confuse them.

Eq.(31)–(35): differential-ring phase noise — with an explicit $N$ (verified ✓)

Original formulas ([P2] Sec. V-B, p.796, verified verbatim against a render of the original PDF):

Power (Eq.(31)) and frequency (Eq.(32)):

$$P=N\,I_{tail}\,V_{DD} \qquad\qquad f_0=\frac{1}{2Nt_D}\approx\frac{1}{2\eta N t_r}\approx\frac{I_{tail}}{2\eta N q_{max}}$$

Noise on each single-ended node (Eq.(33); two shares — differential transistor + load resistor $R_L$ — with $V_{char}=(V_{GS}-V_T)/\gamma$ for a balanced long-channel stage, $E_cL/\gamma$ short-channel):

$$\frac{\overline{i_n^2}}{\Delta f}=\left(\frac{\overline{i_n^2}}{\Delta f}\right)_{N}+\left(\frac{\overline{i_n^2}}{\Delta f}\right)_{Load}=4kT\,I_{tail}\left(\frac{1}{V_{char}}+\frac{1}{R_L I_{tail}}\right)$$

The ring has $2N$ nodes (two outputs per stage), and the total phase noise is $2N$ times the single-source value (p.796, verbatim: "The phase noise and jitter due to all $2N$ noise sources is $2N$ times the value given by (6) and (12)." — this 2 is a node count, not the SSB convention's 2). Combined with the $\Gamma_{rms}$ of Eq.(16), it collapses into (Eq.(34)/(35); $\mathcal{L}$ is the paper's $L\{\Delta f\}$):

$$\mathcal{L}_{min}\{\Delta f\}=\frac{8}{3\eta}\cdot N\cdot\frac{kT}{P}\cdot\left(\frac{V_{DD}}{V_{char}}+\frac{V_{DD}}{R_L I_{tail}}\right)\cdot\frac{f_0^2}{\Delta f^2}$$ $$\kappa_{min}=\sqrt{\frac{8}{3\eta}}\cdot\sqrt{N\cdot\frac{kT}{P}\cdot\left(\frac{V_{DD}}{V_{char}}+\frac{V_{DD}}{R_L I_{tail}}\right)}$$

The paper states verbatim that both are "valid in both long- and short-channel regimes of operation with the right choice of $V_{char}$"; a bipolar differential ring's shot + load noise (Eq.(36), p.797) folds back into the same two equations with $V_{char}=4kT/q_e$.

Meaning: only two differences from single-ended Eq.(23) — an explicit $N$, and an extra load share $V_{DD}/(R_L I_{tail})$ in the bracket. At fixed $f_0$ and $P$, a differential ring's phase noise degrades with $N$ ($\Delta\mathcal{L}=10\log_{10}(N_2/N_1)$; for jitter, $\kappa_{min}\propto\sqrt N$) — the lost other half of N-independence.

Why the $N$ appears (exponent bookkeeping, fixed $P$, $f_0$, fixed swing): $P=NI_{tail}V_{DD}$ is a static bookkeeping (unlike Eq.(21), it is not tied to $f_0$), so fixed $P$ forces $I_{tail}\propto1/N$; Eq.(32) then forces $q_{max}=I_{tail}/(2\eta Nf_0)\propto1/N^2$ (p.797, verbatim: "…reduce the swing, and hence $q_{max}$, by a factor of $1/N^2$"). Hence $2N\cdot\Gamma_{rms}^2\cdot S_i/q_{max}^2\propto N\cdot N^{-3}\cdot N^{-1}\cdot N^{4}=N^{+1}$. Full term-by-term table and worked example: lc_vs_ring Step 2b.

[P2]'s conclusion sentence (pp.796–797, verbatim): "Note that, in contrast with the single-ended ring oscillator, a differential oscillator does exhibit a phase noise and jitter dependency on the number of stages, with the phase noise degrading as the number of stages increases for a given frequency and power dissipation."

Numerical example: $f_0=5$ GHz, $\Delta f=1$ MHz, $kT=4.0\times10^{-21}$ J, $P=1$ mW, $\eta\approx1$, $V_{DD}/V_{char}=3$, $V_{DD}/(R_LI_{tail})=2$ (fixed swing): $N=4$ gives $-82.7$ dBc/Hz, $N=12$ gives $-78.0$ dBc/Hz, $\Delta\mathcal{L}=10\log_{10}3=+4.77$ dB; and $\kappa_{min}$ grows by $\times\sqrt3\approx1.732$. (The absolute values inherit [P2]'s SSB bookkeeping, the same family as the 4 in the denominator of [P1] Eq.(21); the time-domain $/2$ convention shifts everything by $+3$ dB. $\Delta\mathcal{L}$ is a difference, identical under both conventions.)

Python verification:

import numpy as np
def L_ring_diff(N, kT, P, f0, df, eta=1.0, vdd_vchar=3.0, vdd_swing=2.0):  # [P2] Eq.(34)
    return 10*np.log10(8/(3*eta) * N * (kT/P) * (vdd_vchar + vdd_swing) * (f0/df)**2)
L4  = L_ring_diff(4,  4.0e-21, 1e-3, 5e9, 1e6)
L12 = L_ring_diff(12, 4.0e-21, 1e-3, 5e9, 1e6)
print(round(L4,1), round(L12,1), round(L12-L4,2))   # -> -82.7 -78.0 4.77

One-line design rule: single-ended = $N$-free (Eq.23); differential = fewest stages wins (Eq.34, $+3.01$ dB per doubling) — the lower bound on $N$ is set by phase margin / quadrature / multiphase needs, so do not add more. Tail-source noise near $f_0$ "surprisingly" does not enter the phase noise (p.796); what enters is its low-frequency noise (the symmetry path) and its noise near even harmonics (filterable with an LC).

Key figures

Paper figure	Page	Content	Site counterpart	Note
Fig. 5	793	Overlaid ISFs at the same frequency for different stage counts $N$ (3/5/15)	scaling intuition ($\Gamma_{rms}\propto N^{-3/2}$)	✓
Fig. 6	793	Approximate waveform and ISF of one single-ended ring stage (energy concentrated at the transition)	toy triangular ISF (lab_03)	✓
Fig. 8	794	rms ISF vs $N$ for rings of different stage counts; the solid line is Eq.(16) at $\eta=0.75$, $\Gamma_{rms}\approx4/N^{1.5}$	scaling argument for lc_vs_ring_isf_comparison.png	✓
Fig. 9	795	rms ISF vs $N$ for differential rings under three constraint scenarios (fixed power/fixed swing, fixed power/fixed $R_L$, fixed tail current/fixed $R_L$)	empirical support that Eq.(16)'s scaling carries over to differential rings (a premise of the Eq.(34) derivation)	✓
Fig. 17	802	phase noise vs the symmetry (control) voltage, with a minimum at the symmetric point	direct experimental support for the symmetry design rule	✓

Fig. 17 is the smoking gun for the symmetry rule: sweep the control voltage; at the point where the PMOS pull-up current = NMOS pull-down current and the waveform is symmetric, $c_0$ is squeezed to its minimum, 1/f³ upconversion is suppressed, and the phase noise shows a bowl bottom. This directly verifies [P1] Eq.(24) (claim C4).

This site compares LC ($-\sin$) and ring (triangular ISF, peaks shrinking with $N$) with a toy model — not transistor-level:

Design insights

• Jitter and phase noise share one origin: lowering $\Gamma_{rms}^2/q_{max}^2$ lowers both; do not treat long-term jitter and close-in phase noise as two separate problems.
• Adding stages is not a phase-noise cure: for single-ended rings at fixed $f_0$, $P$ the phase noise is nearly independent of $N$ (conclusion verified); differential rings want even fewer stages — Eq.(34) contains an explicit $N$, costing $+3.01$ dB per doubling at fixed $f_0$, $P$. The real reasons to add stages are quadrature/multi-phase outputs, tuning range, and phase margin.
• Symmetry is the master knob for close-in noise: tune the rise/fall to be symmetric (e.g. the control voltage of Fig. 17) to suppress $c_0$ and push the 1/f³ corner far out. Differential rings are usually more symmetric than single-ended ones.
• The steeper the transition, the better: the energy is concentrated at the transitions; the higher the slope, the larger $q_{max}$ and the relatively smaller $\Gamma_{rms}$.

Design-side summaries in lc_vs_ring and symmetry; the SerDes view is in serdes_clocking_connection.

Limitations

Per paper_metadata (paper_002.limitations):

• Toy/first-order: short-channel effects and detailed device noise are approximate.
• The N-independence conclusion holds only for single-ended rings, at fixed power, fixed frequency, and for the specific noise model ([P2] Sec.V, Eq.(23)/(25), p.796, verified, claim C7); a differential ring instead contains an explicit $N$ (Eq.(34), p.796, verified).
• Substrate/supply noise is treated separately and qualitatively.

Relationship to other papers

• [P1] is the foundation: this page's jitter $\kappa$, $\Gamma_{rms}$, and symmetry all use [P1]'s ISF and Eq.(21)/(24).
• [P3]/[P4] also use the ring as a vehicle ([P4]'s ILFD/prescaler is an inverter-chain ring), extending the ISF from phase noise to injection.
• [P5] is unrelated to this page; but latch-based/differential ring start-up also relies on cross-coupled positive feedback (the corner-case bridge of claim C12).

Further reading / companion teaching pages

Which block of this page	Companion teaching page	What that page adds
The LC ($-\sin$) vs ring (transition-concentrated) ISF comparison and the N-scaling argument	lc_vs_ring	The $\Gamma_{rms}$, $q_{max}$, and phase-noise trade-offs of the two topologies organized into a design table
The random walk behind Eq.(8) accumulated jitter $\sigma_{\Delta t}=\kappa\sqrt{\Delta t}$	lab_03_ring_oscillator_toy_model	A runnable toy model: add an independent perturbation per edge; the log-log slope verifies $\sqrt{\Delta t}$ (pedagogical toy, not transistor-level)
The Fig. 17 phase-noise bowl at the symmetric point, $c_0$, and the 1/f³ corner	symmetry	how rise/fall symmetry suppresses $c_0$, differential vs single-ended, design knobs

How to read: this page is the story of "how the paper applies [P1] to the ring"; to watch jitter grow as $\sqrt{\Delta t}$ hands-on, go back to lab_03; to turn the conclusions into topology selection, go back to lc_vs_ring and symmetry. For the SerDes view see also serdes_clocking_connection.

What to remember

• Accumulated jitter $\sigma_{\Delta t}=\kappa\sqrt{\Delta t}$: the random-walk fingerprint of a free-running oscillator ([P2] Eq.(8), p.792).
• $\kappa$ is set by the same $\Gamma_{rms}^2/q_{max}^2$ as phase noise ([P2] Eq.16/23, verified).
• $\Gamma_{rms}\propto N^{-3/2}$ ([P2] Eq.(16), p.794, re-verified in v7: the radical covers only the constant, triple-confirmed by the prose's $4/N^{1.5}$@$\eta=0.75$ and App.B Eq.(55)); yet at fixed $f_0$, $P$ the phase noise is nearly independent of $N$ (no $N$ in [P2] Eq.(23), claim C7, verified).
• Differential rings are the opposite: [P2] Eq.(34), p.796 contains an explicit $N$ ($q_{max}\propto1/N^2$, verbatim on p.797); at fixed $f_0$, $P$ the phase noise degrades as $10\log_{10}N$ ($N=4\to12$ costs $+4.77$ dB); differential designs use the fewest necessary stages.
• Fig. 17: the phase-noise bowl bottom at the symmetric point — the smoking gun for the symmetry rule (claim C4).
• Rings integrate better than LC but usually have worse phase noise; this page shows where the knobs are.