Interactive Tools

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

Seven interactive tools (calculators, explorers, animations, and a sandbox) that turn ISF theory into intuition. Everything computes live in the browser and works offline.

Tool 1: phase noise / jitter calculator

Turn ISF theory into feel: drag the sliders below and watch in real time how the single-sideband phase noise $\mathcal{L}$ (dBc/Hz) and the rms timing jitter $\sigma_t$ (fs) change with the design parameters. Underneath are [P1] Eq.(21) and the 1/f² jitter integral.

ISF 相位雜訊 / jitter 互動計算器

q_max1.0 pC

Γ_rms0.50

S_i = 10^x-24.0 A²/Hz (log)

f₀5.0 GHz

offset (for L)1.0 MHz

integrate from f₁1.00 MHz

integrate to f₂100 MHz

L(offset) — Eq.(21)

-148.0

dBc/Hz

σ_φ (1–100 MHz)

0.06

mrad

σ_t (rms jitter)

1.8

fs

模型：單一白噪源、1/f² skirt（toy）。L = 10·log₁₀[Γ_rms²/q_max² · S_i/(4Δω²)]； σ_t = √(2·L_lin·f_ref²·(1/f₁−1/f₂)) / (2π f₀)。對應 lab_06 / lab_08。

How to play your way to intuition

• Double q_max: $\mathcal{L}$ should drop $6$ dB ($\mathcal{L}\propto 1/q_{max}^2$) — see tank_swing.
• Halve Γ_rms: $\mathcal{L}$ also drops $6$ dB ($\mathcal{L}\propto\Gamma_{rms}^2$) — see symmetry, waveform_slope.
• Pull the lower integration limit f₁ down: $\sigma_t$ grows — because 1/f² jitter is dominated by the lower limit (see Example C in numerical_feeling).
• Raise f₀: at the same $\sigma_\phi$, $\sigma_t=\sigma_\phi/(2\pi f_0)$ shrinks — but do not forget that higher-frequency circuits usually carry more noise too.

Tool 2: ISF Fourier explorer

Tune the ISF Fourier coefficients $c_0$–$c_3$ and watch in real time the waveform $\Gamma(\theta)$, $\Gamma_{rms}$, and the 1/f³ corner ratio $c_0^2/(2\Gamma_{rms}^2)$ ([P1] Eq.24). Pull $c_0$ to 0 (symmetric waveform) and the corner approaches 0 — this is the mechanism that suppresses close-in flicker.

ISF 傅立葉探索器（調 c₀–c₃ 看 Γ(θ) 與 1/f³ corner）

c₀0.00

c₁1.00

c₂0.00

c₃0.00

Γ_rms

0.707

Σcₙ² (=2Γ_rms²)

1.000

1/f³ corner / ω₁f = c₀²/(2Γ_rms²)

0.000

把 c₀→0（對稱波形）→ 1/f³ corner 趨近 0（close-in flicker 被抑制，[P1] Eq.24）。 Γ_rms 決定 1/f² phase noise（[P1] Eq.21）。對應 [fourier](/03_isf_core_theory/fourier_series_of_isf)、[symmetry](/06_design_insights/symmetry)。

Tool 3: SerDes BER bathtub explorer

Take the rms jitter $\sigma_t$ obtained by integrating the oscillator phase noise, together with the UI (inverse baud rate), and look at the eye opening and the BER bathtub. As $\sigma_t$ grows or the UI shrinks, the eye opening at BER $=10^{-12}$ narrows.

SerDes jitter→BER bathtub 探索器（RJ-only）

rms RJ σ_t4.0 ps

UI (1/鮑率)100 ps

eye 開口 @ 1e-12

44.2%

UI（44.2 ps）

σ_t / UI

4.0%

中心 BER

3.7e-36

σ_t 就是把 oscillator phase noise 積分得到的 rms jitter（見 [serdes_clocking_connection](/06_design_insights/serdes_clocking_connection)、[lab_12](/04_simulation_labs/lab_12_serdes_eye_ber)）。 RJ-only toy model（無 ISI/DJ）。

Tool 4: injection-locking explorer

Injection locking (an external signal pulling the oscillator frequency onto itself) has a range set by the ISF fundamental: drag I_inj, the detuning Δω, and the normalized ISF fundamental $\vert\tilde\Gamma_1\vert$ to see the lock range $\omega_L=\tfrac12\,I_{inj}\,\vert\tilde\Gamma_1\vert$ ([P3] Eq.(35)) and the phase dynamics $d\theta/dt=\Delta\omega-\omega_L\sin\theta$ (the Adler equation — external literature, not among the five source PDFs). When $\vert\Delta\omega\vert\le\omega_L$, $\Omega(\theta)=\omega_L\sin\theta$ intersects the horizontal line $\Delta\omega$ and the system locks at a fixed point $\theta^*$; once the detuning exceeds the envelope it unlocks, leaving a residual beat frequency $\Omega_{beat}=\sqrt{\Delta\omega^2-\omega_L^2}$.

Injection Locking 互動探索器（注入鎖定，外部訊號把振盪器頻率拉到自己身上）

I_inj（注入電流，arb.）2.00 a.u.

Δω = ω₀ − ω_inj（detuning）0.40 rad/s

|Γ̃₁|（normalized ISF 基頻）0.50

狀態

LOCKED

|Δω| ≤ ω_L

lock range ω_L — [P3] Eq.(35)

0.500

rad/s

phase error θ*

53.1

deg

讀圖：藍色曲線 Ω(θ)=ω_L·sin(θ) 是「振盪器能提供的最大頻率牽引」； 水平線是要克服的失諧 Δω。只要 Δω 落在 ±ω_L 包絡內， 曲線與水平線就有交點（穩定固定點 = 實心綠點，θ*∈[−90°,+90°]； 另一交點為不穩定），系統鎖定，相位差停在 θ* 不再漂移。 一旦 |Δω| > ω_L 交點消失，相位持續滑動，殘餘拍頻 Ω_beat=√(Δω²−ω_L²)。

公式：lock range ω_L = ½·I_inj·|Γ̃₁|（[P3] Eq.(35)）； 相位動態 dθ/dt = Δω − ω_L·sin(θ)（Adler 方程，外部文獻，非本站 5 篇 PDF）； 鎖定條件 |Δω| ≤ ω_L；拍頻 Ω_beat = √(Δω² − ω_L²)。單位：ω_L、Δω、Ω_beat 同為 rad/s， |Γ̃₁| 無因次，I_inj 為任意單位（normalization 已吸收 q_max）。

Tool 5: Allan deviation explorer

Allan deviation (the two-sample standard deviation that measures long-term time-domain frequency stability) $\sigma_y(\tau)$ translates the same oscillator's noise into a curve versus averaging time $\tau$: pick the dominant FM noise type (white / flicker / random-walk FM) and its strength, and watch the log-log slope fall on the $\tau^{-1/2}$, $\tau^{0}$, $\tau^{+1/2}$ segments in real time. Underneath is the exact closed form of $\sigma_y^2(\tau)=2\int_0^\infty S_y(f)\,\sin^4(\pi f\tau)/(\pi f\tau)^2\,df$ (with $S_y(f)=(f^2/f_0^2)\,S_\phi(f)$) for pure power-law processes, matching the slope table in spec 11.2 and lab_19.

Allan deviation σ_y(τ) 互動探索器

主導 FM 雜訊類型：

level h = 10^x1.0e-20 (1)

mix（混入另兩型）0 %

canonical slope

σ_y ∝ τ^(−1/2)

thermal / shot 白噪 → 1/f² 相位雜訊 → 頻率 random walk-free，sigma_y 隨 τ 下降。

模型：純冪律 FM 過程，S_y(f) = h·f^α（α = 0 / −1 / −2 對應 white / flicker / RW FM）。 Allan 變異數閉式：σ_y² = h/(2τ)（white）、2 ln2·h（flicker）、(2π)²/6·h·τ（RW）， 各為 σ_y²(τ) = 2∫₀^∞ S_y(f)·sin⁴(πfτ)/(πfτ)² df 對純冪律的精確積分結果 （IEEE Std 1139；對應 AUTHORING_SPEC 11.2 與 lab_19）。mix 滑桿把另兩型以變異數相加方式混入 （獨立過程變異數可加），可看出複合曲線在三段斜率間彎折。獨立樣本、無 dead-time 假設。

Tool 6: PLL loop-bandwidth explorer

The loop bandwidth (the cutoff frequency up to which the loop can track the input phase) $f_n$ of a PLL (phase-locked loop) sets the noise budget: the loop low-passes the reference path and high-passes the VCO path, giving the output $S_{out}=S_{ref}\vert H_{lp}\vert^2+S_{vco}\vert H_{hp}\vert^2$ (spec 10.2). Increasing $f_n$ admits more in-band reference noise and less out-of-band VCO noise, so there exists an optimum loop BW that minimizes the integrated jitter $\sigma_t=\sqrt{\int S_{out}\,df}/(2\pi f_0)$ (spec formula 19) — drag the slider to find it. Matches pll_noise_budget and lab_20.

PLL loop-bandwidth 探索器（最佳 loop BW）

loop BW f_n500 kHz

ref noise S_ref = 10^x-12.0 rad²/Hz

VCO noise S_vco@1MHz = 10^x-10.0 rad²/Hz

carrier f₀5.0 GHz

σ_t @ 目前 f_n

476.2

fs（f_n = 500 kHz）

最佳 loop BW f_n*

5.93 MHz

σ_t,min = 197.1 fs

ref/VCO 交越

26 kHz

shaped ref = shaped VCO

離最佳值

142%

σ_t 高於最小

模型：type-II 2nd-order PLL（ζ = 0.707），參考雜訊取平坦 floor、VCO 取 1/f² skirt（@1 MHz 錨定）。低頻被 loop 追蹤（|H_lp|²），高頻 VCO 被高通抑制（|H_hp|²）： S_out = S_ref·|H_lp|² + S_vco·|H_hp|²。積分頻段 1 kHz–1 GHz、σ_t = √(∫S_out df)/(2π f₀) （規範公式 19）。加大 f_n 會多收 in-band 參考雜訊、少收 out-of-band VCO 雜訊，故 存在最小化積分 jitter 的最佳 f_n。對應 pll_utils / lab_13 / lab_20，pedagogical toy model（非特定 silicon loop）。

Tool 7: ISF sandbox — draw a waveform, see the ISF and the phase noise

Use three "shape" sliders (rise/fall slope ratio $A=f'_{rise}/f'_{fall}$, squareness, duty) to draw your own periodic waveform; the sandbox instantly computes the slope-approximation ISF $\Gamma(\theta)$ ([P2] Appendix approach, not the exact PPV), the Fourier coefficients $\vert c_n\vert$ and $\Gamma_{rms}$, then translates them via [P1] Eq.(21) and Eq.(24) into $\mathcal{L}(1\,\text{MHz})$ and the 1/f³ corner. Pull $A$ from 5 back to 1 (more symmetric rise/fall) and watch $c_0\to 0$ with the corner falling all the way to 0 — the hands-on feel of "symmetry↑ → $c_0$↓ → 1/f³ corner↓" from the symmetry page; crank the squareness up (steeper edges) and you get the $\Gamma_{rms}$↓ → $\mathcal{L}$↓ story of waveform_slope.

ISF sandbox: draw a waveform → see the ISF and phase noise

Slope ratio A = f′rise/f′fall1.00

Squareness (0=sine, 1=near-square)0.00

Duty (fraction with V>0)0.50

f₀5.0 GHz

q_max1.00 pC

One period T = 1/f₀ = 200.0 ps; the steepest edge's slew ≈ (dV/dθ)·ω₀ = 31.4 V/ns (amplitude 1 V).

The ISF is a slope approximation ([P2] Appendix approach), not an exact PPV: Γ(θ) = −V′(θ)/f′²_max, with the rising and falling edges each normalized by their own maximum slope (a single global normalization would force c₀ to be identically 0).

Γ_rms

0.707

c₀ (signed)

0.000

L(1 MHz) — [P1] Eq.(21)

-145.0

dBc/Hz

1/f³ corner — [P1] Eq.(24)

0.0

kHz

Model: 512-point numerical computation. L(1 MHz) uses [P1] Eq.(21) (SSB /4 convention) with a fixed S_i = 10⁻²⁴ A²/Hz; 1/f³ corner = c₀²/(2Γ_rms²)·f₁/f, f₁/f = 1 MHz ([P1] Eq.(24)). f₀ only rescales the time axis and the slew (at a fixed offset, Eq.(21) has no explicit f₀ dependence). Anchor point (numerically verified): sine setting (A=1, squareness 0, duty 0.5, q_max=1 pC) → Γ_rms = 0.707, |c₁| = 1.000, c₀ ≈ 0, corner = 0, L(1 MHz) = −145.0 dBc/Hz (Example B's −148.0 dBc/Hz corresponds to Γ_rms = 0.5; 0.707 is 3.0 dB higher than 0.5).

Formula reference

$$\mathcal{L}(\Delta f)=10\log_{10}\!\left(\frac{\Gamma_{rms}^2}{q_{max}^2}\cdot\frac{S_i}{4\,(2\pi\Delta f)^2}\right), \qquad \sigma_t=\frac{1}{2\pi f_0}\sqrt{2\,L_{\text{lin}}\,f_{ref}^2\left(\frac{1}{f_1}-\frac{1}{f_2}\right)}$$

where $S_i=\overline{i_n^2}/\Delta f$ and $L_{\text{lin}}=10^{\mathcal{L}/10}$. This is a single-source, 1/f²-skirt toy model (not transistor-level): real oscillators have multiple noise sources, cyclostationarity, flicker (1/f³), and a noise floor. Full derivations in white_noise_to_phase_noise and psd_phase_noise_jitter.