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

Tank swing, $q_{max}$, and phase noise

Prerequisites: white_noise_to_phase_noise ([P1] Eq.(21) signature result $\mathcal{L}\propto\Gamma_{rms}^2/q_{max}^2$), impulse_to_phase_shift (why $q_{max}=CV_{max}$ lands in the denominator), tank_Q_and_energy_restoration (why high $Q$ is "nearly-free swing", where $R_p$ comes from) | Next: varactor_tuning_supply_pushing, lc_vs_ring

This page answers the question asked earliest — and most often underestimated — in oscillator design: why does making the tank's (LC resonator's) voltage swing larger lower phase noise? The answer is the cleanest scaling in ISF theory: phase noise is inversely proportional to $q_{max}^2$, and $q_{max}=C_{node}V_{max}$ is set directly by swing.

Physical intuition (conclusion first): phase error $=\dfrac{\Gamma}{q_{max}}\Delta q$ — the denominator is $q_{max}$. The larger the swing, the more charge $q_{max}$ the signal carries, and the same lump of noise charge $\Delta q$ becomes proportionally more "negligible" against it, so the phase it can push is smaller. Make the signal big, outweigh the noise — this is the first principle of every low-phase-noise design. The cost: swing is bounded by supply / headroom, and pushing swing up usually burns more power.

Step 1: what $q_{max}$ is, and why it lands in the denominator

$q_{max}$ is "the maximum charge corresponding to the signal swing" at a node:

$$q_{max}=C_{node}\,V_{max}$$

• Unit check: $[\text{F}]\cdot[\text{V}]=[\text{C}]$ ✓.
• It appears in the denominator of the impulse→phase relation ([P1] Eq.(10)–(11), p.182): $\Delta\phi=\dfrac{\Gamma}{q_{max}}\Delta q$.
• Meaning: $\Delta q/q_{max}$ is "the injected charge as a fraction of the signal charge." Larger $q_{max}$ → the same $\Delta q$ produces a smaller relative disturbance → phase is more stable. $\Gamma$ (a dimensionless shape) itself does not change with swing; swing only moves $q_{max}$.

Step 2: the phase-noise $\propto 1/q_{max}^2$ scaling

The signature result for white-noise-induced $1/f^2$ phase noise ([P1] Eq.(21), p.185):

$$\mathcal{L}\{\Delta\omega\}=10\log_{10}\!\left(\frac{\Gamma_{rms}^2}{q_{max}^2}\cdot\frac{\overline{i_n^2}/\Delta f}{4\,\Delta\omega^2}\right)$$

• The denominator has $q_{max}^2$. So $\mathcal{L}\propto1/q_{max}^2$.
• Scaling (claim C3): doubling $q_{max}$ → $\mathcal{L}$ drops by $10\log_{10}(2^2)=6.02$ dB. Every doubling of swing (with $C$ fixed) improves phase noise by 6 dB.

Step-by-step algebra: how "$q_{max}$ doubled → −6 dB" falls out of Eq.(21). Let $q_{max}\to q_{max}'=2q_{max}$, with everything else ($\Gamma_{rms}$, $\overline{i_n^2}/\Delta f$, $\Delta\omega$) unchanged. Phase noise is "dB = $10\log_{10}$(power ratio inside the brackets)", so we only need the new/old ratio inside the brackets:

$$\begin{aligned} \frac{P_{new}}{P_{old}} &=\frac{\dfrac{\Gamma_{rms}^2}{(2q_{max})^2}\cdot\dfrac{\overline{i_n^2}/\Delta f}{4\Delta\omega^2}} {\dfrac{\Gamma_{rms}^2}{q_{max}^2}\cdot\dfrac{\overline{i_n^2}/\Delta f}{4\Delta\omega^2}} \;=\;\frac{1/(2q_{max})^2}{1/q_{max}^2} \;=\;\frac{q_{max}^2}{(2q_{max})^2} \;=\;\frac{1}{4}, \\[6pt] \Delta\mathcal{L}&=\mathcal{L}_{new}-\mathcal{L}_{old} =10\log_{10}\!\left(\frac{P_{new}}{P_{old}}\right) =10\log_{10}\!\left(\frac14\right) =-10\log_{10}4 \\[4pt] &=-20\log_{10}2=-20\times0.30103=-6.02\ \text{dB}. \end{aligned}$$

• What each step uses: line 1 cancels all common factors ($\Gamma_{rms}^2$, $\overline{i_n^2}/\Delta f$, $4\Delta\omega^2$), leaving only the ratio of $q_{max}$; lines 2→3 use the log identity $\log_{10}4=\log_{10}2^2=2\log_{10}2$.
• Why "6 dB/octave" and not 3 dB: because $\mathcal{L}\propto q_{max}^{-2}$ is an inverse-square law — a power ratio of $\tfrac14$ corresponds to $-6$ dB (not $-3$ dB; $-3$ dB is $\tfrac12$). Doubling voltage/charge → power ratio $4\times$ → $20\log_{10}2$.
• Dimension check: the ratio $P_{new}/P_{old}$ is dimensionless (same units cancel) → $10\log_{10}$ gives dB (dimensionless) ✓.
• Dimension check (the whole bracket must be dimensionless to give dBc/Hz): $\dfrac{(\text{dimensionless})^2}{[\text{C}]^2}\cdot\dfrac{[\text{A}^2/\text{Hz}]}{[\text{rad/s}]^2} =\dfrac{[\text{A}^2/\text{Hz}]}{[\text{C}^2][\text{s}^{-2}]}$. Using $[\text{A}]=[\text{C/s}]$: $=\dfrac{[\text{C}^2\text{s}^{-2}/\text{Hz}]}{[\text{C}^2\text{s}^{-2}]}=\dfrac{1}{[\text{Hz}]}$. $\overline{i_n^2}/\Delta f$ is itself already per-Hz, and it is exactly this per-Hz factor that makes the absolute $\mathcal{L}$ per-Hz (dBc/Hz); and (as above) the ratio $P_{new}/P_{old}$ itself is dimensionless, so $10\log_{10}$ gives dB ✓.

The same $1/q_{max}^2$ also appears in the ring oscillator's accumulated-jitter proportionality constant ([P2] Eq.(11)–(12), p.793):

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

• So "larger swing lowers phase noise" and "larger swing lowers jitter" are the same statement — they share the core ratio $\Gamma_{rms}^2/q_{max}^2$. ([P2] Eq.(12), p.793 verified verbatim: $\kappa=\dfrac{\Gamma_{rms}}{q_{max}}\sqrt{\tfrac12\,\overline{i_n^2}/\Delta f}$.)

Step 3: two routes to raise $q_{max}$ — increase $V_{max}$ vs. increase $C$

$q_{max}=C\cdot V_{max}$, so in principle either raising $C$ or raising $V_{max}$ raises $q_{max}$. But the two routes have completely different costs:

Route	Change in $q_{max}$	Effect on phase noise	Side effect
Increase $V_{max}$ (swing)	$q_{max}\propto V_{max}$	$\mathcal{L}\propto1/V_{max}^2$ (most effective)	Limited by supply / breakdown / headroom; needs more bias current to sustain swing
Increase $C$ (tank capacitance)	$q_{max}\propto C$	Looks like $\propto1/C^2$, but there is a trap ↓	To hold $f_0$, $L$ must drop proportionally; sustaining the same swing needs a larger tank current ($Q$, $g_m$ limited); $f_0=1/\sqrt{LC}$ is tied down

• Increasing swing is usually the first choice: direct, 6 dB/octave, and does not move $f_0$.
• Increasing $C$ has a trap: in an LC tank $f_0=1/(2\pi\sqrt{LC})$, so raising $C$ requires lowering $L$ proportionally; sustaining the same swing voltage then needs more tank current ($Q$, $g_m$ limited), and the real-world improvement is often eaten by "the extra power/noise spent driving the larger $C$." The genuinely clean lever is pushing swing to the headroom limit within the power budget.

Step 4: power / voltage headroom trade-off

Swing cannot grow without bound — it hits two walls, supply and device:

1. Voltage headroom: the single-ended swing limit of a differential LC tank ranges from about the supply $V_{DD}$ (current-limited regime) up to $\sim\dfrac{4}{\pi}\,I_{bias}R_p$ (where $R_p$ is the tank's equivalent parallel resistance). Once pushed into the voltage-limited regime, adding more current no longer adds swing — the phase-noise improvement saturates.
2. Power: in the current-limited regime, swing $\approx\dfrac{4}{\pi}I_{bias}R_p$ is proportional to bias current. Doubling swing → doubling current → doubling power.

Putting the two together gives the FOM (figure of merit) trade-off:

• phase noise $\mathcal{L}\propto1/q_{max}^2\propto1/V_{max}^2$ (bigger swing is always better),
• but $V_{max}\propto I_{bias}$ (current-limited) → $\mathcal{L}\propto1/I_{bias}^2$, while $P\propto I_{bias}$,
• so does $\mathcal{L}\propto1/P^2$? — No. Because the noise current $\overline{i_n^2}$ also rises with bias current (more current → more device noise), typically $\overline{i_n^2}\propto I_{bias}$, and the net effect reverts to $\mathcal{L}\cdot P\approx$ constant (roughly 1 dB of phase noise bought per extra 1 dB of power burned).
• This is why the industry uses FOM $=\mathcal{L}-20\log_{10}(f_0/\Delta f)+10\log_{10}(P/1\text{mW})$ to fairly compare oscillators at different power levels: it normalizes away this "phase noise × power ≈ constant" trade-off.

⚠️ The "current-limited / voltage-limited" boundary above and $\dfrac{4}{\pi}I_{bias}R_p$ are standard LC-oscillator design knowledge (not among the five downloaded source PDFs; supplemented from standard literature, e.g. the Hajimiri-Lee textbook, Razavi's RF Microelectronics). [P1] itself gives the $1/q_{max}^2$ scaling but does not expand the circuit-level detail of swing-vs-power. The exact swing-limit coefficients and phase-noise-factor floor can be found in E. Hegazi, H. Sjöland, A. A. Abidi, A Filtering Technique to Lower LC Oscillator Phase Noise, IEEE JSSC 36(12):1921–1930, 2001 (verified verbatim), and Razavi's RF Microelectronics (external literature, not among the five source PDFs).

Numerical example (building intuition)

Using canonical example B as the baseline, we look at the effect of doubling swing.

Baseline (example B): $f_0=5$ GHz, $\Delta f=1$ MHz, $q_{max}=1$ pC, $\Gamma_{rms}=0.5$, $S_i=10^{-24}$ A²/Hz:

$$\mathcal{L}=10\log_{10}\!\left(\frac{0.25}{(10^{-12})^2}\cdot\frac{10^{-24}}{4(2\pi\cdot10^6)^2}\right).$$

First $\Delta\omega=2\pi\cdot10^6=6.283\times10^6$ rad/s, $\Delta\omega^2=3.948\times10^{13}$. Bracket $=\dfrac{0.25}{10^{-24}}\cdot\dfrac{10^{-24}}{4\cdot3.948\times10^{13}}=\dfrac{0.25}{1.579\times10^{14}}=1.583\times10^{-15}$, $\mathcal{L}=10\log_{10}(1.583\times10^{-15})=-148.0$ dBc/Hz.

Swing doubled ($q_{max}=2$ pC, everything else unchanged): $q_{max}^2$ becomes 4× → bracket becomes 1/4 →

$$\mathcal{L}_{new}=-148.0-10\log_{10}(4)=-148.0-6.02=-154.0\ \text{dBc/Hz}.$$

• Intuition: doubling swing → phase noise improves by 6.0 dB, exactly $20\log_{10}2$.
• Trade-off reminder: but if this 6 dB comes from doubling bias current (current-limited, with $\overline{i_n^2}\propto I$), $S_i$ also rises by ~3 dB, so the net improvement is only about 3 dB — this is "phase noise × power ≈ constant" at work. The genuine free lunch is "raise $V_{max}$ without adding current" (e.g., a higher-$Q$ tank, higher $R_p$).

Design knobs to lower phase noise / raise $q_{max}$ (checklist)

Knob	Affects	Mechanism	Cost / notes
Increase voltage swing $V_{max}$	$q_{max}\uparrow$	$\mathcal{L}\propto1/V_{max}^2$, 6 dB/octave	Limited by headroom / breakdown
Raise tank $Q$ (lower loss, $R_p\uparrow$)	$V_{max}\uparrow$ (more swing at same current)	Higher $R_p$ → same $I_{bias}$ gives more swing, free noise reduction	Limited by process inductor $Q$, parasitics
Differential topology	Effective swing ×2	Differential swing is twice single-ended → $q_{max}\uparrow$	Double the devices/area/power
Push bias to the current/voltage boundary	Maximize $V_{max}$	Take all available headroom	Past the voltage-limited point it saturates — only wastes current
Lower $\Gamma_{rms}$ (the other lever)	$\mathcal{L}\propto\Gamma_{rms}^2$	Equally important as $q_{max}$	See device_noise_mapping

Note: $\mathcal{L}\propto\Gamma_{rms}^2/q_{max}^2$ — $q_{max}$ and $\Gamma_{rms}$ are two independent levers. This page covers $q_{max}$ (swing); $\Gamma_{rms}$ (waveform shape, cyclostationary) is covered in device_noise_mapping.

Validity and failure conditions

Condition	When it holds	When it fails
Small perturbation, ISF unchanged by swing	$\mathcal{L}\propto1/q_{max}^2$ holds cleanly	Once swing is large enough to change waveform shape/$\Gamma$, the scaling deviates
Current-limited regime	Swing $\propto I_{bias}$, buys phase noise	After voltage-limited, more current is useless
$\overline{i_n^2}$ unchanged by swing	Full 6 dB/octave is realized	If noise rises with bias, the net improvement is discounted

Worked examples

The following two problems use [P1] Eq.(21) step by step to compute "swing/$q_{max}$ change → $\mathcal{L}$ change," continuing with canonical example B: $f_0=5$ GHz, $\Delta f=1$ MHz, $\Gamma_{rms}=0.5$, $S_i=\overline{i_n^2}/\Delta f=10^{-24}$ A²/Hz.

Example 1 ($q_{max}$ doubled → $\mathcal{L}$ drops 6 dB, step-by-step with Eq.(21)) Baseline $q_{max}=1$ pC gives $\mathcal{L}=-148.0$ dBc/Hz (example B). Double the swing so $q_{max}=2$ pC ($C$ fixed, $V_{max}$ doubled), everything else unchanged. Find the new $\mathcal{L}$.

Step-by-step substitution (with units), plugging directly into Eq.(21) for the absolute value (no approximation), then checking against −6 dB:

$$\begin{aligned} \Delta\omega&=2\pi\Delta f=2\pi\times10^{6}=6.283\times10^{6}\ \text{rad/s},\quad \Delta\omega^2=3.948\times10^{13}\ \text{(rad/s)}^2, \\[4pt] \text{bracket}_{new}&=\frac{\Gamma_{rms}^2}{(q_{max}')^2}\cdot\frac{S_i}{4\Delta\omega^2} =\frac{(0.5)^2}{(2\times10^{-12}\,\text{C})^2}\cdot\frac{10^{-24}\,\text{A}^2/\text{Hz}}{4\times3.948\times10^{13}} \\[4pt] &=\frac{0.25}{4\times10^{-24}}\cdot\frac{10^{-24}}{1.579\times10^{14}} =\frac{0.25}{4\times1.579\times10^{14}}=3.958\times10^{-16}, \\[4pt] \mathcal{L}_{new}&=10\log_{10}(3.958\times10^{-16})=-154.0\ \text{dBc/Hz}. \end{aligned}$$

• Result: $\mathcal{L}_{new}=-154.0$ dBc/Hz, exactly 6.0 dB lower than the baseline $-148.0$ — matching the $-20\log_{10}2$ algebraic conclusion above.
• Dimension check: inside the bracket, $\dfrac{\text{dimensionless}}{[\text{C}]^2}\cdot\dfrac{[\text{A}^2/\text{Hz}]}{[\text{rad/s}]^2}$, using $[\text{A}]=[\text{C/s}]$ → $[\text{A}^2]=[\text{C}^2\text{s}^{-2}]$, numerator $[\text{C}^2\text{s}^{-2}/\text{Hz}]$, denominator $[\text{C}^2\text{s}^{-2}]$ → leaves $1/[\text{Hz}]$, absorbed into per-Hz → dimensionless power ratio, $10\log_{10}$ gives dBc/Hz ✓.
• One-line Python check:

import numpy as np
def L_eq21(grms, qmax, Si, dw):
    return 10*np.log10(grms**2/qmax**2 * Si/(4*dw**2))
dw = 2*np.pi*1e6
L0 = L_eq21(0.5, 1e-12, 1e-24, dw)   # baseline
L1 = L_eq21(0.5, 2e-12, 1e-24, dw)   # qmax doubled
print(round(L0,1), round(L1,1), round(L1-L0,2))  # -> -148.0 -154.0 -6.02

Example 2 (moving two levers at once: swing doubled + $\Gamma_{rms}$ halved) Starting from the baseline ($q_{max}=1$ pC, $\Gamma_{rms}=0.5$), double the swing ($q_{max}=2$ pC) and improve the waveform so $\Gamma_{rms}=0.25$ (halved). Find the total improvement $\Delta\mathcal{L}$.

Step-by-step substitution. Since $\mathcal{L}\propto\Gamma_{rms}^2/q_{max}^2$, the two levers add in the log domain:

$$\begin{aligned} \Delta\mathcal{L}&=10\log_{10}\!\left(\frac{(\Gamma_{rms}')^2/(q_{max}')^2}{\Gamma_{rms}^2/q_{max}^2}\right) =\underbrace{10\log_{10}\!\left(\frac{(0.25)^2}{(0.5)^2}\right)}_{\Gamma_{rms}\,\text{halved}} +\underbrace{10\log_{10}\!\left(\frac{(10^{-12})^2}{(2\times10^{-12})^2}\right)}_{q_{max}\,\text{doubled}} \\[4pt] &=10\log_{10}(0.25)+10\log_{10}(0.25)=(-6.02)+(-6.02)=-12.04\ \text{dB}. \end{aligned}$$

• Result: total improvement −12 dB (each lever contributes −6 dB). Relative to example B, $-148.0$ → $-160.0$ dBc/Hz. This shows $\Gamma_{rms}$ and $q_{max}$ are two independent, additive levers (the numerator and denominator of [P1] Eq.(21)).
• Dimension check: both ratios are dimensionless → dB values still add as dB ✓.
• One-line Python check:

import numpy as np
print(round(L_eq21(0.25, 2e-12, 1e-24, 2*np.pi*1e6) - L_eq21(0.5, 1e-12, 1e-24, 2*np.pi*1e6), 2))
# -> -12.04 dB   (reuse L_eq21 from Example 1)

Reminder (see Step 4): if these improvements come from bias current (current-limited, $\overline{i_n^2}\propto I_{bias}$), $S_i$ rises along with it and the net improvement is discounted — this is "phase noise × power ≈ constant" again. The two problems above assume $S_i$ fixed (the ideal upper bound).

Key takeaways

• $q_{max}=C\cdot V_{max}$; phase noise $\mathcal{L}\propto1/q_{max}^2$ ([P1] Eq.(21)).
• Doubling $q_{max}$ → $\mathcal{L}$ improves by 6.02 dB (e.g., $-148\to-154$ dBc/Hz).
• Increasing swing is cleaner than increasing $C$ (increasing $C$ ties down $f_0$ and needs more current); raising tank $Q$ is nearly-free swing.
• Trade-off: in the current-limited regime, swing $\propto$ current, and device noise $\propto$ current too → phase noise × power ≈ constant; use FOM for a fair comparison.
• $q_{max}$ and $\Gamma_{rms}$ are two independent levers (the numerator and denominator of [P1] Eq.(21)).

Further reading

• Signature-formula derivation: white_noise_to_phase_noise
• The role of $q_{max}$ in impulse→phase: impulse_to_phase_shift
• Where tank $Q$ comes from, why high $Q$ equals free swing, $R_p$ and $4kT/R_p$ thermal noise: tank_Q_and_energy_restoration
• How tuning-line/supply jitter FMs the carrier (another phase-noise gateway): varactor_tuning_supply_pushing
• The other lever, $\Gamma_{rms}$: device_noise_mapping
• Connection to slope/swing: waveform_slope
• LC vs. ring swing differences: lc_vs_ring