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

Lab 09 — Design trade-off scaling synthesis

Breadcrumb: Simulation labs › Noise & jitter › This page (design trade-offs intro). Upstream: lab_06, lab_08; advanced version: lab_17.

The first eight labs exercised each mechanism one at a time. This synthesis lab introduces no new simulation; instead it treats [P1] Eq.(21) — the signature equation — as a map of design knobs: holding everything else fixed, separately sweep $q_{max}$, $\Gamma_{rms}$, and the ring stage count $N$, and watch how the phase noise $\mathcal{L}$ and the rms jitter $\sigma_t$ move. Throughout we use the canonical numbers (spec Section 8) for back-of-envelope arithmetic, building the reflex of "turn this knob, gain that many dB".

Physical intuition (conclusion first): $1/f^2$ phase noise $\propto\Gamma_{rms}^2/q_{max}^2$. There are only two broad roads to lower phase noise: make the signal charge $q_{max}$ larger (store more energy, so the noise is relatively smaller), or make the ISF $\Gamma_{rms}$ smaller (make the waveform insensitive to noise). Both are $20\log_{10}$ relationships: doubling $q_{max}$ → phase noise $-6$ dB; halving $\Gamma_{rms}$ → also $-6$ dB. A ring oscillator's $N$ moves two things at once — watch the net effect carefully.

1. Learning objectives

• Read [P1] Eq.(21) as a scaling law and build design intuition for $q_{max}$, $\Gamma_{rms}$, and $N$.
• Use the canonical numbers to compute "how many dB / how many fs one knob buys".
• Understand the two-sided nature of the ring oscillator's $N$ ($\Gamma_{rms}\propto N^{-3/2}$, but power/area grow too).
• Connect the scaling conclusions to the design-insight pages.

2. Mathematical model

The starting point is [P1] Eq.(21), p.185:

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

Expand the logarithm into "additive knobs" (fixing $\Delta\omega$ and $S_i=\overline{i_n^2}/\Delta f$):

$$\mathcal{L}=20\log_{10}\Gamma_{rms}-20\log_{10}q_{max}+10\log_{10}S_i-20\log_{10}(2\Delta\omega)+\text{const}.$$

• How to read it: every knob enters as $20\log_{10}$ (voltage/charge-like) or $10\log_{10}$ (power-like). Both $q_{max}$ and $\Gamma_{rms}$ enter through $20\log_{10}$, so "double/halve = $\pm6$ dB".

Jitter and phase noise share the same origin: for the $1/f^2$ skirt, $\sigma_t=\frac{1}{2\pi f_0}\sqrt{\int S_\phi df}$ (spec formula 19), and $S_\phi\propto\Gamma_{rms}^2/q_{max}^2$, so

$$\sigma_t\;\propto\;\frac{\Gamma_{rms}}{q_{max}}\qquad(\text{固定}f_0,\ S_i,\ \text{積分範圍}).$$

• The bridge from dB to jitter: $\mathcal{L}$ down $6$ dB (power $\times\frac14$) → $\sigma_t$ halves (the $\sqrt{}$).

Ring oscillator: [P2] gives two scalings — more stages make each stage's ISF sharper but its rms smaller ([P2] Eq.(16), p.794):

$$\Gamma_{rms}=\sqrt{\dfrac{2\pi^2}{3\eta^3}}\;\dfrac{1}{N^{1.5}}\quad(\text{[P2] Eq.(16), p.794 — v7 re-verified: the square root covers only the constant, }\Gamma_{rms}\propto N^{-3/2}\text{; }\approx4/N^{1.5}\text{ at η=0.75, the solid line in [P2] Fig.8}),\qquad f_0=\frac{1}{2N\tau_D}\ \text{[P2] Eq.(15)}.$$

• Looking at phase noise alone, $N$ from 5 → 15 ($\times3$): $\Gamma_{rms}$ is multiplied by $3^{-3/2}=0.192$, a $\mathcal{L}$ change of $20\log_{10}0.192=-14.3$ dB. But more stages → power, area, and the $q_{max}$ allocation all change, and $f_0$ drops (unless $\tau_D$ shrinks), so this is a toy scaling that "looks at $\Gamma_{rms}$ in isolation", not a net design conclusion.

3. Block diagram

4. Core Python code

This lab produces no new figure; the scaling comes straight from back-of-envelope use of [P1] Eq.(21). The snippet below (following the conventions of simulations/lab_08_jitter_integration.py) shows how to turn knob changes into dB and fs so you can sweep the parameters yourself:

import math

def L_dbc(Grms, qmax, Si, df):
    """Hajimiri-Lee Eq.(21): 1/f^2 SSB phase noise [dBc/Hz]."""
    dw = 2 * math.pi * df
    return 10 * math.log10((Grms**2 / qmax**2) * (Si / (4 * dw**2)))

# canonical baseline (spec example B)
f0, df, Si = 5e9, 1e6, 1e-24
base = L_dbc(Grms=0.5, qmax=1e-12, Si=Si, df=df)   # -148.0 dBc/Hz

print("baseline           :", round(base, 1), "dBc/Hz")
print("q_max x2           :", round(L_dbc(0.5, 2e-12, Si, df) - base, 1), "dB")  # -6.0
print("Gamma_rms /2       :", round(L_dbc(0.25, 1e-12, Si, df) - base, 1), "dB") # -6.0
# ring N: Gamma_rms ~ N^-1.5 ([P2] Eq.(16), p.794, re-verified v7: sqrt covers only the constant); show isolated scaling
for N in (5, 15, 45):
    rel = (N / 5.0) ** -1.5
    print(f"N={N:2d} Gamma_rms rel={rel:.3f}  dL={20*math.log10(rel):+.1f} dB")

# jitter scales as Gamma_rms / q_max  ->  -6 dB phase noise = sigma_t / 2

• L_dbc is not the same thing as lab_08's integrate_rms_jitter: lab_08 integrates jitter out of $\mathcal{L}$; here we go the other way — use Eq.(21) to generate $\mathcal{L}$, then infer jitter from the scaling. The two pages are forward and inverse operations of each other.

5. Full script path

This lab is a synthesis analysis with no dedicated script. The calculations it cites come from: simulations/lab_06_white_noise_phase_noise.py (simulation evidence for the $\Gamma_{rms}^2/q_{max}^2$ scaling), simulations/lab_03_ring_toy_model.py (LC vs ring ISF and the $N$ scaling), simulations/lab_08_jitter_integration.py ($\mathcal{L}\to\sigma_t$). The back-of-envelope snippet above can be saved as a small standalone script for your own parameter sweeps.

6. Parameter table

Parameter	Symbol	Canonical value	Role
Carrier frequency	$f_0$	$5$ GHz	Fixed
Offset	$\Delta f$	$1$ MHz	Fixed (evaluation point)
Current noise PSD	$S_i$	$1\times10^{-24}$ A²/Hz	Fixed (single white source)
Maximum charge swing	$q_{max}$	$1$ pC (baseline)	Knob 1, swept over $0.5,1,2$ pC
ISF rms	$\Gamma_{rms}$	$0.5$ (baseline)	Knob 2, swept over $0.25,0.5$
Ring stage count	$N$	$5$ (baseline)	Knob 3, swept over $5,15,45$
Baseline phase noise	$\mathcal{L}$	$-148.0$ dBc/Hz	From Eq.(21) (= spec example B)

7. Unit table

Quantity	Symbol	Unit
Maximum charge swing	$q_{max}$	C
ISF rms	$\Gamma_{rms}$	Dimensionless
Ring stage count	$N$	Dimensionless
Phase noise	$\mathcal{L}$	dBc/Hz
rms jitter	$\sigma_t$	s
Current noise PSD	$S_i$	A²/Hz

8. Scaling table (back-of-envelope, toy scaling)

Take the canonical baseline $\mathcal{L}=-148.0$ dBc/Hz ($q_{max}=1$ pC, $\Gamma_{rms}=0.5$, $S_i=10^{-24}$, 5 GHz @ 1 MHz) as the $0$ dB reference. $\sigma_t\propto\Gamma_{rms}/q_{max}$.

Knob change	$\mathcal{L}$ change	$\sigma_t$ change	Physical reason
$q_{max}\times2$ ($\to2$ pC)	$-6.0$ dB	$\times0.5$	$-20\log_{10}q_{max}$; larger signal charge, relatively smaller noise
$q_{max}\times0.5$ ($\to0.5$ pC)	$+6.0$ dB	$\times2$	Same as above, reversed
$\Gamma_{rms}\times0.5$ ($\to0.25$)	$-6.0$ dB	$\times0.5$	$+20\log_{10}\Gamma_{rms}$; waveform insensitive to noise
Ring $N:5\to15$ ($\times3$)	$-14.3$ dB	$\times0.192$	$\Gamma_{rms}\propto N^{-3/2}$ (in isolation; [P2] Eq.(16), p.794, v7 re-verified)
Ring $N:5\to45$ ($\times9$)	$-28.6$ dB	$\times0.037$	Same as above; but power/area/$f_0$ change too

Back-of-envelope example: doubling $q_{max}$ from 1 pC to 2 pC changes $\mathcal{L}$ by $20\log_{10}(1/2)=-6.0$ dB → $-154.0$ dBc/Hz; jitter halves via the $\sqrt{}$. Apply the same $-6$ dB move to lab_08's $447.9$ fs (that was the $-100$ dBc/Hz scenario) and it drops to $\approx224$ fs.

9. How to read the figures (reusing existing figures)

Figure 1: simulation evidence for the $\Gamma_{rms}^2/q_{max}^2$ scaling. In lab_06's white-noise PSD figure, swapping the ISF for a different $\Gamma_{rms}$ (or changing $q_{max}$) merely shifts the whole $1/f^2$ line up or down, without changing its slope — the visual version of the scaling table's "turn a knob = shift by dB".

Figure 2: LC vs ring ISF shape and $N$. lab_03's comparison figure: the ring's ISF sensitivity is concentrated at the transitions; the larger $N$, the sharper each stage's ISF but the smaller its rms ($\Gamma_{rms}\propto N^{-3/2}$). This explains where knob 3 in the scaling table comes from — and reminds you it is an isolated scaling.

• How to use the two figures: Figure 1 tells you "the $q_{max}$/$\Gamma_{rms}$ knobs = a vertical shift"; Figure 2 tells you "how the ring's $\Gamma_{rms}$ varies with $N$". Together they form the design map.

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: [P2] Eq.(23), p.796, $\mathcal{L}|_{1/f^2}\approx\frac{8}{3\eta}\,\frac{V_{DD}}{V_{char}}\,\frac{kT}{P}(\omega_0/\Delta\omega)^2$ ($\eta$ is the stage-delay proportionality constant of [P2] Eq.(14), $\approx1$; $\gamma$ enters only through $V_{char}=\Delta V/\gamma$).
• Concept figures: reuses white_noise_phase_noise_psd.png (lab_06) and lc_vs_ring_isf_comparison.png (lab_03).

11. Limitations and approximations

• Toy scaling, not transistor-level: the scaling assumes "turn one knob, hold everything else exactly fixed". In a real circuit $q_{max}$, $\Gamma_{rms}$, $N$, power, area, and $f_0$ are coupled and cannot be tuned in isolation.
• Ring $N$ 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 $N^{-3/4}$): the proportionality constant is $\Gamma_{rms}=\sqrt{2\pi^2/(3\eta^3)}\cdot\dfrac{1}{N^{1.5}}$ ($\eta\approx1$); adding stages also changes $f_0=1/(2N\tau_D)$, power, and area — the net phase noise/jitter requires the full FOM, not $\Gamma_{rms}$ alone.
• Single white source: ignores multiple sources, cyclostationarity ($\Gamma_{eff}=\Gamma\cdot\alpha$), and flicker upconversion ($1/f^3$, see lab_07). The table does not apply in the close-in region.
• Factor-of-2: uses Eq.(21)'s denominator $4\Delta\omega^2$; it differs from the time-domain version by the factor-of-2 SSB bookkeeping, which affects none of the scaling conclusions (all dB changes are relative).
• Jitter scaling: $\sigma_t\propto\Gamma_{rms}/q_{max}$ assumes a fixed integration range and a $1/f^2$ shape; if close-in $1/f^3$ dominates, jitter is set by $c_0$ and the lower integration limit and must be computed separately.

Key takeaways

• $1/f^2$ phase noise $\propto\Gamma_{rms}^2/q_{max}^2$; jitter $\propto\Gamma_{rms}/q_{max}$.
• Doubling $q_{max}$ or halving $\Gamma_{rms}$ → phase noise $-6$ dB, jitter halved ($20\log_{10}$).
• Ring $N$ in isolation: $\Gamma_{rms}\propto N^{-3/2}$ ($N\times3\to-14.3$ dB), but $f_0$/power/area change simultaneously — a toy scaling.
• Baseline numbers: $q_{max}=1$ pC, $\Gamma_{rms}=0.5$, 5 GHz @ 1 MHz, $S_i=10^{-24}$ → $\mathcal{L}=-148.0$ dBc/Hz.

Further reading

• $q_{max}$ design: power_and_qmax
• Symmetry cuts $1/f^3$: symmetry_and_1overf
• SerDes clocking: serdes_clocking_connection
• Applied to design/theory: turning the $\Gamma_{rms}$, $q_{max}$, $N$ scalings into a topology decision → lc_vs_ring
• Upstream simulations: lab_06_white_noise_phase_noise, lab_08_jitter_integration