Lab 17 — Design Sweeps: Three Design Curves for swing / Γrms / N

Breadcrumb: Simulation labs › System & advanced › This page (advanced design sweeps). Back-of-the-envelope intro version: lab_09; upstream: lab_06, lab_08.

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

lab_09 read [P1] Eq.(21) as a scaling law by mental arithmetic. This lab actually sweeps the same equation and plots it as three curves: holding everything else fixed, sweep the node charge swing $q_{max}$ (swing), the ISF rms $\Gamma_{rms}$ , and the ring stage count $N$ , and watch how the phase noise $\mathcal{L}$ at $1$ MHz offset moves. On the plots you will see with your own eyes the " $-20$ dB/decade" straight-line slope, the " $2\times$ swing $\to-6$ dB" arrow, and the ring oscillator's nearly horizontal $N$ curve (the famous $N$ -independence).

Physical intuition (conclusion first): $1/f^2$ phase noise $\propto\Gamma_{rms}^2/q_{max}^2$ . Both knobs enter as $20\log_{10}$ : doubling $q_{max}$ → $-6$ dB; halving $\Gamma_{rms}$ → $-6$ dB. The ring's $N$ is sneakier: adding stages shrinks $\Gamma_{rms}\propto N^{-3/2}$ , but the per-node $q_{max}$ shrinks and there are more noisy devices — the three roughly cancel at fixed $f_0$ /power, so net phase noise is almost independent of $N$ .

1. Learning objectives

Plot [P1] Eq.(21)'s $\mathcal{L}\propto\Gamma_{rms}^2/q_{max}^2$ as three design curves.
On the $q_{max}$ curve, quantify " $-20$ dB/decade" and " $2\times$ swing $\to-6$ dB".
On the $\Gamma_{rms}$ curve, see "symmetric waveform / low $\Gamma_{rms}\to$ PN↓".
On the ring $N$ curve, see where $N$ -independence comes from ([P2] Eq.(16), p.794 — v7 re-verified: $\Gamma_{rms}\propto N^{-3/2}$ ).
Connect the curves back to lab_09 and the Chapter 06 design-insight pages.

2. Mathematical model

The starting point is again [P1] Eq.(21), p.185 ( $1/f^2$ region, fixed offset):

\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)

Expand the logarithm to expose each knob's own slope:

\mathcal{L}=20\log_{10}\Gamma_{rms}-20\log_{10}q_{max}+10\log_{10}\!\big(\overline{i_n^2}/\Delta f\big)-20\log_{10}(2\Delta\omega)+\text{const}.

Knob 1 ( $q_{max}$ ): $-20\log_{10}q_{max}$ → against a $\log$ horizontal axis in $q_{max}$ this is a straight line with slope $-20$ dB/decade; $q_{max}\times2\Rightarrow20\log_{10}2=6.02$ dB → $-6$ dB.
Knob 2 ( $\Gamma_{rms}$ ): $+20\log_{10}\Gamma_{rms}$ → halving $\Gamma_{rms}$ → $-6$ dB; this lab uses a linear horizontal axis, so the curve is the logarithmic rise of $20\log_{10}\Gamma_{rms}$ .

The ring's $N$ ([P2] Eq.(16), p.794 — v7 re-verified: the square root covers only the constant, $\Gamma_{rms}\propto N^{-3/2}$ ; the body text's $4/N^{1.5}$ at η=0.75 and App. B Eq.(55) triple-confirm this. v3 had misread it as $N^{-3/4}$ ): adding stages moves three things at once, and the three cancel each other at fixed $f_0$ /power:

\Gamma_{rms}\propto N^{-3/2}\ \text{[P2] Eq.(16)},\qquad q_{max}\propto N^{-1}\ (\text{smaller per-node swing}),\qquad \overline{i_n^2}\propto N\ (\text{more devices}).

Substituting into $\Gamma_{rms}^2/q_{max}^2\cdot\overline{i_n^2}$ :

\frac{(N^{-3/2})^2}{(N^{-1})^2}\cdot N=\frac{N^{-3}}{N^{-2}}\cdot N=N^{-1}\cdot N=N^{0}=\text{const}.

Conclusion: under this toy scaling the net effect $\propto N^0$ → phase noise is independent of $N$ ([P2]'s signature result).
Dimension check: $\Gamma_{rms}^2/q_{max}^2\cdot(\text{A}^2/\text{Hz})/(\text{rad/s})^2$ → the bracket is dimensionless (before taking $10\log_{10}$ ), legitimate ✓; $N$ is dimensionless throughout ✓.

Numbers for this lab: evaluated at $\Delta f=1$ MHz, $\overline{i_n^2}/\Delta f=10^{-22}$ A²/Hz; the baseline $q_{max}=1$ pC, $\Gamma_{rms}=0.5$ gives $\mathcal{L}=-128.0$ dBc/Hz.

3. Block diagram

4. Core Python code

simulations/lab_17_design_sweep.py uses a shared L_dbc to evaluate [P1] Eq.(21) in dBc/Hz, then sweeps one curve per knob:

import numpy as np


def L_dbc(Grms, qmax, in2_df, dw):
    return 10 * np.log10(Grms ** 2 / qmax ** 2 * in2_df / (4 * dw ** 2))


dw = 2 * np.pi * 1e6        # evaluate at 1 MHz offset
in2_df = 1e-22

# (a) sweep q_max: 0.1 pC .. 10 pC  -> slope -20 dB/decade, 2x -> -6 dB
q = np.logspace(-13, -11, 50)
L_q = L_dbc(0.5, q, in2_df, dw)                # baseline qmax=1pC -> -128 dBc/Hz

# (b) sweep Gamma_rms: low Grms -> lower PN
g = np.linspace(0.1, 1.5, 50)
L_g = L_dbc(g, 1e-12, in2_df, dw)

# (c) ring N: Gamma_rms ~ N^-3/2, per-node qmax ~ 1/N, more noisy devices ~ N
N = np.arange(3, 31)
Grms_N = 0.5 * (5.0 / N) ** 1.5                # scaling (illustrative; [P2])
qmax_N = 1e-12 * (5.0 / N)                      # lower per-node swing as N grows
noise_N = in2_df * (N / 5.0)                    # more noisy devices
L_N = 10 * np.log10(Grms_N ** 2 / qmax_N ** 2 * noise_N / (4 * dw ** 2))
print(round(L_N[0], 1), round(L_N[-1], 1))
# -> -128.0 -128.0  (essentially flat for all N: the famous N-independence)

L_dbc vs lab_08: lab_08 integrates the measured $\mathcal{L}$ into jitter; here we go the other way, using Eq.(21) to generate $\mathcal{L}$ and then sweep the knobs. The two pages are inverse operations of each other (same note as in lab_09).
The three scalings in (c) are illustrative: the constant in $\Gamma_{rms}\propto N^{-3/2}$ and the exponents for per-node $q_{max}$ and device count are toy assumptions, meant to draw concretely "why $N$ roughly cancels".

5. Full script path

simulations/lab_17_design_sweep.py (main() draws the three subplots; L_dbc wraps [P1] Eq.(21) in dBc/Hz; (c) uses the toy scaling $\Gamma_{rms}\propto N^{-3/2}$ , $q_{max}\propto N^{-1}$ , $\overline{i_n^2}\propto N$ ). Re-run: python scripts/run_all_sims.py.

Note: this page's filename is lab_17_design_tradeoffs.md (aligned with the sidebar naming convention); the corresponding script is lab_17_design_sweep.py, and the figure file is design_tradeoff_sweeps.png.

6. Parameter table

Parameter	Symbol	Value / sweep range	Role
Offset (evaluation point)	$\Delta f$	$1$ MHz	Fixed
Current noise PSD	$\overline{i_n^2}/\Delta f$	$10^{-22}$ A²/Hz	Fixed (baseline); varies with $N$ in (c)
Maximum charge swing	$q_{max}$	(a) $0.1$ – $10$ pC; baseline $1$ pC	Knob 1
ISF rms	$\Gamma_{rms}$	(b) $0.1$ – $1.5$ ; baseline $0.5$	Knob 2
Ring stage count	$N$	(c) $3$ – $30$ ; baseline $5$	Knob 3
Baseline phase noise	$\mathcal{L}$	$-128.0$ dBc/Hz	From Eq.(21) ( $q_{max}=1$ pC, $\Gamma_{rms}=0.5$ )

7. Unit table

Quantity	Symbol	Unit
Maximum charge swing	$q_{max}$	C
ISF rms	$\Gamma_{rms}$	Dimensionless
Ring stage count	$N$	Dimensionless
Offset frequency	$\Delta f,\ \Delta\omega$	Hz, rad/s
Phase noise	$\mathcal{L}$	dBc/Hz
Current noise PSD	$\overline{i_n^2}/\Delta f$	A²/Hz

8. Simulation figure

Three design curves: (a) q_max sweep at -20dB/dec, (b) Γrms sweep, (c) ring N nearly horizontal

9. How to read the figure

(a) swing sweep (blue): on a log horizontal axis, $q_{max}$ (0.1–10 pC) vs $\mathcal{L}$ is a straight line with slope $-20$ dB/decade — as $q_{max}$ goes from 0.1 → 10 pC (two decades), $\mathcal{L}$ drops from $-108$ → $-148$ dBc/Hz, exactly $40$ dB. The red arrow marks " $2\times$ swing $\to-6$ dB" ( $1\to2$ pC, $-128\to-134$ dBc/Hz). Mnemonic: double the swing, gain 6 dB of phase noise.
(b) $\Gamma_{rms}$ sweep (green): linear horizontal axis, a logarithmic curve $\mathcal{L}=20\log_{10}\Gamma_{rms}+$ const — smaller $\Gamma_{rms}$ means lower phase noise, with the steepest descent at the small- $\Gamma_{rms}$ end (halving $\Gamma_{rms}$ from 0.5 to 0.25 → $-6$ dB). This is the payoff of "symmetric waveform / low sensitivity".
(c) ring $N$ sweep (purple): a nearly perfectly horizontal line ( $\approx-128$ dBc/Hz, independent of $N$ ). Not a coincidence — it is the $N^0$ computed in Section 2: more stages shrink $\Gamma_{rms}$ (good), but $q_{max}$ also shrinks and there are more devices (bad); the three cancel at fixed $f_0$ /power. Adding ring stages does not buy you phase noise for free ([P2]).
Putting it together: (a)(b) are the "knobs that truly earn dB" (raise the swing, suppress $\Gamma_{rms}$ ); (c) is the "looks tunable, actually cancels" knob — choose $N$ based on $f_0$ , power, area, and phase-count requirements, not for phase noise.

10. Corresponding paper equations/figures

Main scaling: [P1] Eq.(21), p.185, $\mathcal{L}\propto\Gamma_{rms}^2/q_{max}^2$ .
Parseval / $\Gamma_{rms}$ : [P1] Eq.(20), p.185.
Ring $\Gamma_{rms}$ scaling: [P2] Eq.(16), p.794 (v7 re-verified: the square root covers only the constant, $\Gamma_{rms}\propto N^{-3/2}$ ; the body text's $4/N^{1.5}$ at η=0.75 and App. B Eq.(55) triple-confirm this. v3 had misread it as $N^{-3/4}$ ).
Ring frequency: [P2] Eq.(15), p.794, $f_0=1/(2N\tau_D)$ .
Ring white-noise FOM / $N$ -independence: [P2], p.795, $\mathcal{L}\vert_{1/f^2}\approx\frac{8}{3\eta}\,\frac{V_{DD}}{V_{char}}\,\frac{kT}{P}(\omega_0/\Delta\omega)^2$ (the prefactor of [P2] Eq.(23), p.796 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 this to $8/(3\gamma)$ and mislabeled it "verified verbatim"; v3 corrected it against the original PDF p.796).

11. Limitations and approximations

Toy scaling, not transistor-level: (a)(b) assume "turn one knob only, everything else perfectly fixed"; in real circuits $q_{max}$ , $\Gamma_{rms}$ , $N$ , power, area, and $f_0$ are coupled and cannot be tuned in isolation.
The three exponents in ring (c) are illustrative: $\Gamma_{rms}\propto N^{-3/2}$ , $q_{max}\propto N^{-1}$ , $\overline{i_n^2}\propto N$ are toy assumptions to demonstrate the cancellation; the exact constants are marked TODO: manual verification needed from [P2] page 794–796. Changing $N$ also changes $f_0=1/(2N\tau_D)$ (unless $\tau_D$ is shrunk), which this figure does not show simultaneously.
$1/f^2$ region only, fixed offset: evaluated at $1$ MHz; close-in $1/f^3$ is dominated by $c_0$ and the lower integration limit, where these three plots do not apply (see lab_07).
Single white noise source: ignores multiple sources, cyclostationarity ( $\Gamma_{eff}=\Gamma\alpha$ , lab_14), and AM–PM.
Factor-of-2: uses Eq.(21)'s $4\Delta\omega^2$ convention; the factor-of-2 SSB-bookkeeping difference does not affect any relative (dB) slope.
Jitter inference: $\sigma_t\propto\Gamma_{rms}/q_{max}$ ( $-6$ dB phase noise = jitter halved) requires a fixed integration range and $1/f^2$ shape; see lab_08, lab_09.

Key takeaways

$q_{max}$ curve: $-20$ dB/decade; $2\times$ swing $\to-6$ dB (baseline $-128$ → $-134$ dBc/Hz).
$\Gamma_{rms}$ curve: $20\log_{10}\Gamma_{rms}$ , halving $\to-6$ dB; symmetric waveforms / low sensitivity pay off.
Ring $N$ curve: nearly horizontal — $\Gamma_{rms}^2/q_{max}^2\cdot\overline{i_n^2}\propto N^0$ , phase noise is independent of $N$ ([P2] Eq.(16), p.794, v7 re-verified: $\Gamma_{rms}\propto N^{-3/2}$ ).
Baseline: $q_{max}=1$ pC, $\Gamma_{rms}=0.5$ , $\overline{i_n^2}/\Delta f=10^{-22}$ , $1$ MHz $\to\mathcal{L}=-128.0$ dBc/Hz.

1. Learning objectives​

2. Mathematical model​

3. Block diagram​

4. Core Python code​

5. Full script path​

6. Parameter table​

7. Unit table​

8. Simulation figure​

9. How to read the figure​

10. Corresponding paper equations/figures​

11. Limitations and approximations​

Key takeaways​

Further reading​