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公式推導索引 Equation Index

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#ConceptFinal Formula推導頁 Derivation來源 SourceNotes
1impulse chargeΔq=∫i(t) dt\Delta q = \int i(t)\, dtimpulse_to_phase_shiftpaper_001 (general.pdf), around Eq. (9), p.182step
2charge to voltage stepΔV=ΔqCnode\Delta V = \dfrac{\Delta q}{C_{node}}impulse_to_phase_shiftpaper_001, Eq. (9), p.182step
3impulse-to-phase (ISF definition in operational form)Δϕ=Γ(ω0τ)qmax Δq\Delta\phi = \dfrac{\Gamma(\omega_0\tau)}{q_{max}}\,\Delta qimpulse_to_phase_shiftpaper_001, from Eq. (10)-(11), p.182final
4excess-phase impulse response (LTV)hϕ(t,τ)=Γ(ω0τ)qmax u(t−τ)h_\phi(t,\tau) = \dfrac{\Gamma(\omega_0\tau)}{q_{max}}\,u(t-\tau)isf_definitionpaper_001, Eq. (10), p.182final
5LTV phase response (convolution)ϕ(t)=1qmax∫−∞tΓ(ω0τ) in(τ) dτ\phi(t) = \dfrac{1}{q_{max}}\int_{-\infty}^{t}\Gamma(\omega_0\tau)\,i_n(\tau)\,d\tauconvolution_derivationpaper_001, Eq. (11), p.182final
6ISF Fourier seriesΓ(ω0τ)=c02+∑n=1∞cncos⁡(nω0τ+θn)\Gamma(\omega_0\tau) = \dfrac{c_0}{2} + \sum_{n=1}^{\infty} c_n\cos(n\omega_0\tau+\theta_n)fourier_series_of_isfpaper_001, Eq. (12), p.183final
7rms ISF (Parseval)∑n=0∞cn2=1π∫02π∣Γ(x)∣2dx=2 Γrms2\sum_{n=0}^{\infty} c_n^2 = \dfrac{1}{\pi}\int_0^{2\pi}\vert \Gamma(x)\vert ^2 dx = 2\,\Gamma_{rms}^2rms_isfpaper_001, Eq. (20), p.185final
8white noise phase noise (1/f^2)L{Δω}=10log⁡10 ⁣(Γrms2qmax2⋅in2‾/Δf4 Δω2)\mathcal{L}\{\Delta\omega\} = 10\log_{10}\!\left(\dfrac{\Gamma_{rms}^2}{q_{max}^2}\cdot\dfrac{\overline{i_n^2}/\Delta f}{4\,\Delta\omega^2}\right)white_noise_to_phase_noisepaper_001, Eq. (21), p.185final
9flicker noise upconversion (1/f^3)L{Δω}=10log⁡10 ⁣(c02qmax2⋅in2‾/Δf8 Δω2⋅ω1/fΔω)\mathcal{L}\{\Delta\omega\} = 10\log_{10}\!\left(\dfrac{c_0^2}{q_{max}^2}\cdot\dfrac{\overline{i_n^2}/\Delta f}{8\,\Delta\omega^2}\cdot\dfrac{\omega_{1/f}}{\Delta\omega}\right)flicker_noise_upconversionpaper_001, Eq. (23), p.185final
101/f^3 cornerΔω1/f3=ω1/f⋅c022 Γrms2\Delta\omega_{1/f^3} = \omega_{1/f}\cdot\dfrac{c_0^2}{2\,\Gamma_{rms}^2}flicker_noise_upconversionpaper_001, Eq. (24), p.185final
11SSB phase noise vs phase PSDL(Δf)≈12Sϕ(Δf)\mathcal{L}(\Delta f) \approx \tfrac{1}{2} S_\phi(\Delta f)psd_phase_noise_jitterstandard small-angle relation; consistent with paper_001 usagefinal
12phase error to timing errorΔt=Δϕ2πf0\Delta t = \dfrac{\Delta\phi}{2\pi f_0}psd_phase_noise_jitterstandard; used throughout paper_002 and SerDes practicefinal
13phase variance from PSDσϕ2=∫f1f2Sϕ(f) df\sigma_\phi^2 = \int_{f_1}^{f_2} S_\phi(f)\, dfpsd_phase_noise_jitterstandardfinal
14rms jitter from phase varianceσt=σϕ2πf0=12πf0∫f1f2Sϕ(f) df\sigma_t = \dfrac{\sigma_\phi}{2\pi f_0} = \dfrac{1}{2\pi f_0}\sqrt{\int_{f_1}^{f_2} S_\phi(f)\, df}serdes_clocking_connectionstandard; SerDes clockingfinal
15accumulated (ring) jitter random walkσΔt=κ Δt\sigma_{\Delta t} = \kappa\,\sqrt{\Delta t}lab_03_ring_oscillator_toy_modelpaper_002, Eq. (8), p.792 (kappa via Eq.(12), p.793)final
16ring frequency vs stagesf0=12NτDf_0 = \dfrac{1}{2 N \tau_D}lc_vs_ringpaper_002, Eq. (14), p.794final
17ring rms ISF scalingΓrms=2π23η3  1N1.5  ⇒  Γrms∝N−3/2 (Γrms2∝N−3)\Gamma_{rms}=\sqrt{\dfrac{2\pi^2}{3\eta^3}}\;\dfrac{1}{N^{1.5}}\;\Rightarrow\;\Gamma_{rms}\propto N^{-3/2}\ (\Gamma_{rms}^2\propto N^{-3})（η=0.75\eta=0.75 時 ≈4/N1.5\approx 4/N^{1.5}，即 [P2] Fig.8 的實線；根號只含常數）lc_vs_ring[P2] Eq.(16), p.794（v7 已重核：根號只蓋常數，Γrms∝N−3/2\Gamma_{rms}\propto N^{-3/2}；正文 4/N1.54/N^{1.5}@η=0.75\eta=0.75 與 App.B Eq.(55) 三重驗證。v3 曾誤讀為 N−3/4N^{-3/4}）final (verified)
18ring phase noise FOM (white)L{Δf}=83η⋅kTP⋅VDDVchar⋅(f0Δf)2(min at VT=0: 16γ3η)\mathcal{L}\{\Delta f\}=\dfrac{8}{3\eta}\cdot\dfrac{kT}{P}\cdot\dfrac{V_{DD}}{V_{char}}\cdot\left(\dfrac{f_0}{\Delta f}\right)^2\quad(\text{min at }V_T=0:\ \tfrac{16\gamma}{3\eta})lc_vs_ringpaper_002, Eq. (23),(25), p.796。前置係數為 8/(3η)8/(3\eta)（η\eta 為級延遲比例常數 Eq.14，≈1\approx 1）；γ\gamma 僅透過 Vchar=ΔV/γV_{char}=\Delta V/\gamma 進入。（v2 曾誤改為 8/(3γ)8/(3\gamma) 並誤標「逐字核實」，v3 已對照原始 PDF p.796 更正。）final (verified)
19Leeson model (comparison only)L(Δω)=10log⁡10 ⁣[2FkTPs(1+(ω02QΔω)2)(1+ω1/f3∣Δω∣)]\mathcal{L}(\Delta\omega) = 10\log_{10}\!\left[\dfrac{2FkT}{P_s}\left(1+\left(\dfrac{\omega_0}{2Q\Delta\omega}\right)^2\right)\left(1+\dfrac{\omega_{1/f^3}}{\vert \Delta\omega\vert }\right)\right]equation_indexLeeson (1966), Proc. IEEE 54(2):329-330, DOI 10.1109/PROC.1966.4682 (citation verified; external, not in the 5 PDFs). Discussed in paper_001 intro.reference
20generalized Adler / phase equation (injection)dθdt=(ω0−ωinj)+Ω(θ),Ω(θ)=1Tinj∫TinjΓ~(ωinjt+θ) iinj(t) dt,  Γ~=Γ/qmax\frac{d\theta}{dt}=(\omega_0-\omega_{inj})+\Omega(\theta),\quad \Omega(\theta)=\frac{1}{T_{inj}}\int_{T_{inj}}\tilde\Gamma(\omega_{inj} t+\theta)\,i_{inj}(t)\,dt,\ \ \tilde\Gamma=\Gamma/q_{max}paper_003_injection_locking_part1paper_003 (Hong Part I 2019)：Γ~=Γ/qmax\tilde\Gamma=\Gamma/q_{max} Eq.(26)；時間平均之廣義 Adler 方程 Eq.(30), p.2113（原文平均項取 ＋ 號，平均週期 TinjT_{inj}）；鎖定範圍 ωL=12Iinj∣Γ~1∣\omega_L=\tfrac{1}{2} I_{inj}\vert\tilde\Gamma_1\vert Eq.(35), p.2114。本站 Ω(θ)\Omega(\theta) 與差頻 (ω0−ωinj)(\omega_0-\omega_{inj}) 同號相加，符號慣例與 [P3] 一致。final (verified)
21amplitude perturbation function (APF)Λ~ (APF): d(t,ϕ)=e−t/τ0, τ0=2Qωosc;Γ~1=1qmax∠90∘, Λ~1=τ0qmax∠0∘ (quadrature)\tilde\Lambda\ (\text{APF}):\ d(t,\phi)=e^{-t/\tau_0},\ \tau_0=\frac{2Q}{\omega_{osc}};\quad \tilde\Gamma_1=\tfrac{1}{q_{max}}\angle90^\circ,\ \tilde\Lambda_1=\tfrac{\tau_0}{q_{max}}\angle0^\circ\ (\text{quadrature})paper_004_injection_locking_part2paper_004 (Hong Part II 2019), Eq.(25),(26),(27), p.2128 (verified verbatim)final (verified)
22PPV / adjoint (broader literature, NOT in the 5 PDFs)ϕ˙(t)=v1T(t) B(t) ξ(t)(Demir et al. PPV form)\dot{\phi}(t) = v_1^T(t)\, B(t)\, \xi(t)\quad\text{(Demir et al. PPV form)}effective_isfDemir-Mehrotra-Roychowdhury (2000), IEEE TCAS-I 47(5):655-674, DOI 10.1109/81.847872 (citation verified; external, not in the 5 PDFs).reference
23ring kappa (jitter rate)κ=Γrmsqmax12in2‾Δf(σΔt=κΔt)\kappa=\dfrac{\Gamma_{rms}}{q_{max}}\sqrt{\dfrac{1}{2}\dfrac{\overline{i_n^2}}{\Delta f}}\quad(\sigma_{\Delta t}=\kappa\sqrt{\Delta t})lab_03_ring_oscillator_toy_modelpaper_002, Eq. (12), p.793 (verified verbatim)final (verified)
24ring per-stage device noisein2‾Δf=4kTγ gd0=4kTγ μCoxWL ΔV\dfrac{\overline{i_n^2}}{\Delta f}=4kT\gamma\,g_{d0}=4kT\gamma\,\mu C_{ox}\dfrac{W}{L}\,\Delta Vdevice_noise_mappingpaper_002, Eq. (17),(18), p.795 (verified verbatim)final (verified)
25Lorentzian carrier lineshapeS(Δω)∝DD2+Δω2,Δf3dB=Dπ,D=Γrms24qmax2in2‾Δf=κ22S(\Delta\omega)\propto\dfrac{D}{D^2+\Delta\omega^2},\quad \Delta f_{3\mathrm{dB}}=\dfrac{D}{\pi},\quad D=\dfrac{\Gamma_{rms}^2}{4q_{max}^2}\dfrac{\overline{i_n^2}}{\Delta f}=\dfrac{\kappa^2}{2}lorentzian_linewidthphase random walk -> exponential carrier autocorrelation -> Lorentzian; resolves 1/f^2 divergence at Df->0. ISF map corrected v5 (was /(2q^2); MC-adjudicated, lab_23). Background: Demir 2000 [E2].final (verified)
26Allan variance from PSDσy2(τ)=2∫0∞Sy(f)sin⁡4(πfτ)(πfτ)2 df,Sy=f2f02Sϕ\sigma_y^2(\tau)=2\int_0^\infty S_y(f)\dfrac{\sin^4(\pi f\tau)}{(\pi f\tau)^2}\,df,\quad S_y=\dfrac{f^2}{f_0^2}S_\phiallan_variancewhite/flicker/RW FM -> sigma_y ~ tau^-0.5, tau^0, tau^0.5. External: Allan 1966 [E1-ext].final
27PLL output phase-noise budgetSout=(SrefN2+Scp)∣Hlp∣2+Svco∣Hhp∣2S_{out}=(S_{ref}N^2+S_{cp})\lvert H_{lp}\rvert^2+S_{vco}\lvert H_{hp}\rvert^2pll_noise_budgetsum of all PLL noise sources shaped by their transfer; optimal loop BW minimizes integrated jitter.final
28ISF as harmonic transfer vector (HTM)input at f→output at f+kf0 with gain ck (ISF Fourier coeff)\text{input at } f \to \text{output at } f+kf_0 \text{ with gain } c_k\ (\text{ISF Fourier coeff})ltv_htmrigorous LTV / harmonic transfer matrix view; ISF is the phase output's HTM row. External: Zadeh 1950 [E5].reference

v5–v8 新增公式（jitter 核、擴散字典、非對稱閉式、[P3]/[P4] 進階、ring 相關雜訊）​

本節整合 v5–v8 開發波次新增的 headline 公式；欄位與上方主表一致，另加「站內出處」標明教學頁內的推導章節。每條逐字讀自對應教學頁（不重新推導），教學頁本身已對照論文 PDF 核實。

#ConceptFinal Formula推導頁 Derivation來源 Source站內出處Notes
29timing error = sampled phase (TIE kernel)σTIE2=1ω02∫f1f2Sϕ(f) df\sigma_{\text{TIE}}^2=\dfrac{1}{\omega_0^2}\displaystyle\int_{f_1}^{f_2}S_\phi(f)\,dfjitter_kernels本站推導；與 paper_002 Eq.(46)–(49), p.803（Khinchin 路線）同構第 3 步「核 (a)：TIE」final
30N-period jitter kernel 4sin⁡2(πfNT)4\sin^2(\pi fNT)σP2(N)=1ω02∫0∞Sϕ(f) 4sin⁡2(πfNT) df\sigma_P^2(N)=\dfrac{1}{\omega_0^2}\displaystyle\int_0^\infty S_\phi(f)\,4\sin^2(\pi fNT)\,dfjitter_kernels本站推導；核與 paper_002 Eq.(49), p.803（單邊 4sin⁡24\sin^2，經雙邊換算逐字核實）等價第 3 步「核 (b)：N-period」final (verified)
31cycle-to-cycle jitter kernel 16sin⁡4(πfT)16\sin^4(\pi fT)σc2c2=1ω02∫0∞Sϕ(f) 16sin⁡4(πfT) df\sigma_{c2c}^2=\dfrac{1}{\omega_0^2}\displaystyle\int_0^\infty S_\phi(f)\,16\sin^4(\pi fT)\,dfjitter_kernels本站推導；paper_002 Eq.(51), p.803（cycle-to-cycle，本站區分兩種定義）第 3 步「核 (c)：cycle-to-cycle」final (verified)
32white-FM closed form (核積分 = [P2] 隨機漫步)σΔϕ2(N)=κ2 NT(κ2=Γrms2Si/(2qmax2))\sigma_{\Delta\phi}^2(N)=\kappa^2\,NT\quad(\kappa^2=\Gamma_{rms}^2 S_i/(2q_{max}^2))jitter_kernelspaper_002, Eq.(8)/(11)/(12), p.792–793（已核實）；本站頻域核積分精確回收第 4 步「白噪 FM 封閉式」final (verified)
33[P2] Eq.(45)–(51), p.803（jitter ← phase spectrum，自相關/Khinchin）Rϕ(τ)=∫−∞∞Sϕ(f)ej2πfτdfR_\phi(\tau)=\displaystyle\int_{-\infty}^{\infty}S_\phi(f)e^{j2\pi f\tau}df（雙邊，Eq.48）；σΔϕ2=8ω02∫0∞Sϕ(f)sin⁡2(πfτ) df\sigma_{\Delta\phi}^2=\dfrac{8}{\omega_0^2}\displaystyle\int_0^\infty S_\phi(f)\sin^2(\pi f\tau)\,df（Eq.49）；κ=Δff010−L{Δf}/20\kappa=\dfrac{\Delta f}{f_0}10^{-\mathcal{L}\{\Delta f\}/20}（Eq.50）jitter_kernelspaper_002, Eq.(45)–(51), p.803（v5 逐字核實，見「與哪些 paper/公式對應」節）「與哪些 paper／公式對應」節final (verified)
34diffusion dictionary 主角 κ2\kappa^2（[P2] Eq.11）κ2≡Γrms22qmax2⋅in2‾Δf [rad2/s]\kappa^2\equiv\dfrac{\Gamma_{rms}^2}{2q_{max}^2}\cdot\dfrac{\overline{i_n^2}}{\Delta f}\ [\text{rad}^2/\text{s}]（canonical 0.1250.125）diffusion_dictionarypaper_002, Eq.(11), p.793（逐字核實）第 0 步「主角只有一個」final (verified)
35D=κ2/2D=\kappa^2/2（慣例乙，本站規範採用）D=Γrms24qmax2in2‾Δf=κ22D=\dfrac{\Gamma_{rms}^2}{4q_{max}^2}\dfrac{\overline{i_n^2}}{\Delta f}=\dfrac{\kappa^2}{2}（canonical 0.06250.0625）diffusion_dictionary對應 Demir 2000 慣例（外部文獻）；v5 修正（v3 曾誤用慣例甲之值 D=κ2D=\kappa^2）「衣服二：D」節final (verified)
36Lorentzian FWHM =κ2/2π=\kappa^2/2\pi（=D/π，v5 映射）Δf3dB=κ22π=Dπ\Delta f_{3\mathrm{dB}}=\dfrac{\kappa^2}{2\pi}=\dfrac{D}{\pi}（canonical 19.919.9 mHz；真 LC 39.839.8 mHz）diffusion_dictionary衍生自 paper_002 Eq.(11) 的 κ2\kappa^2；外部文獻 Demir 2000 [E2]（Lorentzian 機制）「衣服三：Lorentzian 3-dB 線寬」節final (verified)
37b−2b_{-2}：1/f² phase PSD 係數Sϕ(f)=2κ2(2πf)2≡b−2f2 [rad2/Hz]， b−2=κ22π2S_\phi(f)=\dfrac{2\kappa^2}{(2\pi f)^2}\equiv\dfrac{b_{-2}}{f^2}\ [\text{rad}^2/\text{Hz}]，\ b_{-2}=\dfrac{\kappa^2}{2\pi^2}（canonical 6.33×10−36.33\times10^{-3}）diffusion_dictionary與 paper_001 Eq.(21), p.185（/4/4 版）、white_noise_to_phase_noise 時域推導一致「衣服四：1/f² phase PSD 係數」節final (verified)
38App.B Γrms2(N,A)\Gamma_{rms}^2(N,A) 閉式（非對稱三角 ISF）Γrms2=2π23η31N3[41+A3(1+A)3]\Gamma_{rms}^2=\dfrac{2\pi^2}{3\eta^3}\dfrac{1}{N^3}\Big[4\dfrac{1+A^3}{(1+A)^3}\Big]（A=1A=1 精確退化為 Eq.16）asymmetric_isf_closed_formpaper_002, App. B, Eq.(52)–(55), p.803（v8 本次渲染逐字核實）第 1–3 步final (verified)
39App.B Γdc(N,A)\Gamma_{dc}(N,A) 閉式（DC 值 / c0c_0 來源）Γdc=2πη21N2(1−A1+A)(c0=2Γdc)\Gamma_{dc}=\dfrac{2\pi}{\eta^2}\dfrac{1}{N^2}\Big(\dfrac{1-A}{1+A}\Big)\quad(c_0=2\Gamma_{dc})asymmetric_isf_closed_formpaper_002, App. B, Eq.(56), p.803（v8 本次渲染逐字核實）第 4 步final (verified)
40App.B 1/f³ corner 閉式（N,AN,A 直接算）f1/f3=f1/f⋅32ηN⋅(1−A)21−A+A2f_{1/f^3}=f_{1/f}\cdot\dfrac{3}{2\eta N}\cdot\dfrac{(1-A)^2}{1-A+A^2}（[P1] Eq.24 慣例值 =2×=2\times 本式）asymmetric_isf_closed_formpaper_002, App. B, Eq.(57), p.803（v8 本次渲染逐字核實；對應主文 Eq.7, p.792）第 5 步final (verified)
41[P1] Appendix Eq.(31) 切向投影l=ΔX⃗⋅X⃗˙∣X⃗˙∣l=\Delta\vec X\cdot\dfrac{\dot{\vec X}}{\lvert\dot{\vec X}\rvert}isf_from_waveformpaper_001, Appendix, Eq.(31), p.192（逐字轉錄）方法 B 第 1 步final (verified)
42[P1] Appendix Eq.(32)–(33)：位移→時間→相位Δt=l∣X⃗˙∣=ΔX⃗⋅X⃗˙∣X⃗˙∣2\Delta t=\dfrac{l}{\lvert\dot{\vec X}\rvert}=\Delta\vec X\cdot\dfrac{\dot{\vec X}}{\lvert\dot{\vec X}\rvert^2}（Eq.32）；Δϕ=2πTΔt\Delta\phi=\dfrac{2\pi}{T}\Delta t（Eq.33）isf_from_waveformpaper_001, Appendix, Eq.(32)–(33), p.193（逐字轉錄）方法 B 第 2–3 步final (verified)
43[P1] Appendix Eq.(34)–(36)：節點電壓特例、closed-form ISFΔϕi=2πTΔqiCiv˙i∣v⃗˙∣2\Delta\phi_i=\dfrac{2\pi}{T}\dfrac{\Delta q_i}{C_i}\dfrac{\dot v_i}{\lvert\dot{\vec v}\rvert^2}（Eq.34）；Γi(x)=fi′∑jfj′ 2\Gamma_i(x)=\dfrac{f_i'}{\sum_j f_j'^{\,2}}（Eq.36）isf_from_waveformpaper_001, Appendix, Eq.(34)–(36), p.193（逐字轉錄）方法 B 第 4–5 步final (verified)
44[P1] Appendix Eq.(37)：二階系統 closed form（f=cos⁡x⇒Γ=−sin⁡xf=\cos x\Rightarrow\Gamma=-\sin x）Γ(x)=f′f′ 2+f′′ 2\Gamma(x)=\dfrac{f'}{f'^{\,2}+f''^{\,2}}isf_from_waveformpaper_001, Appendix, Eq.(37), p.193（逐字轉錄；正弦自我檢驗核實）方法 B 第 6 步final (verified)
45[P1] Appendix Eq.(38)：一階導數近似（ring 專用）Γi(x)=fi′(x)fmax′ 2\Gamma_i(x)=\dfrac{f_i'(x)}{f_{max}'^{\,2}}isf_from_waveformpaper_001, Appendix, Eq.(38), p.193（逐字轉錄）方法 Cfinal (verified)
46[P3] Sec. IV 脈衝列 Eq.(19)–(21)：kick 與頻移Δϕ=±qinjqmax\Delta\phi=\pm\dfrac{q_{inj}}{q_{max}}（Eq.19）；Δω=ΔϕTinj\Delta\omega=\dfrac{\Delta\phi}{T_{inj}}（Eq.20–21）paper_003_injection_locking_part1paper_003 (Hong Part I 2019), Sec. IV, Eq.(19)–(21), p.2112（已核實）「脈衝列鎖定」節 Step 1–2final (verified)
47[P3] Eq.(22)–(23)：能量平衡與脈衝列基波振幅ω0qmax=Q Iosc\omega_0 q_{max}=Q\,I_{osc}（Eq.22）；Iinj=2qinjTinjI_{inj}=\dfrac{2q_{inj}}{T_{inj}}（Eq.23）paper_003_injection_locking_part1paper_003, Sec. IV, Eq.(22)–(23), p.2112（已核實）「脈衝列鎖定」節 Step 4final (verified)
48Lab 36 / [P3] Eq.(38)–(40)：捕獲暫態、pull-in frequencyωc≡ωLcos⁡θss=ωL2−Δω2\omega_c\equiv\omega_L\cos\theta_{ss}=\sqrt{\omega_L^2-\Delta\omega^2}（[P3] Eq.40 的 ωp\omega_p）；精確解 R(θ(t))=R(θ0)e−ωctR(\theta(t))=R(\theta_0)e^{-\omega_c t}lab_36_lock_acquisitionpaper_003, Eq.(38)–(40), p.2115（已核實）；本站精確閉式解為站內推導，與 [P4] Eq.(31)–(32) tanh 形式等價2.1 節「Part (a) 捕獲暫態」final (verified)
49Lab 36：傾斜搓衣板障壁與 Kramers slip 率（外部文獻）ΔU+=2ωL[1−r2−rarccos⁡r]\Delta U_+=2\omega_L\big[\sqrt{1-r^2}-r\arccos r\big]；νslip≈ωc2πe−ΔU+/D\nu_{slip}\approx\dfrac{\omega_c}{2\pi}e^{-\Delta U_+/D}lab_36_lock_acquisition障壁為本站推導（自 U(θ)=−Δωθ−ωLcos⁡θU(\theta)=-\Delta\omega\theta-\omega_L\cos\theta）；逃逸率為 Kramers 1940、Risken 1989（外部文獻，非本站 5 篇 PDF）2.2 節「Part (b) 傾斜搓衣板」reference
50[P3] Eq.(43)–(45)：最佳注入波形與 lock range 上限（Cauchy–Schwarz）Irms≡⟨iinj2⟩I_{rms}\equiv\sqrt{\langle i_{inj}^2\rangle}（Eq.43）；iinj,0∗(x)=±IrmsΓ~rmsΓ~(x)i_{inj,0}^{*}(x)=\pm\dfrac{I_{rms}}{\tilde\Gamma_{rms}}\tilde\Gamma(x)（Eq.44）；ωL∗=IrmsΓ~rms\omega_L^*=I_{rms}\tilde\Gamma_{rms}（Eq.45）injection_locking_noisepaper_003, Sec. VI, Eq.(43)–(45), pp.2119–2120（逐字核實）「注入波形設計」節 第 0–3 步final (verified)
51[P4] Eq.(28)–(30)：M:N 次諧波鎖定 lock characteristicφ(t)≡MNωinjt+θ(t)\varphi(t)\equiv\dfrac{M}{N}\omega_{inj}t+\theta(t)（Eq.28）；dθdt=ω0−MNωinj+1NTinj∫NTinj ⁣ ⁣Γ~(MNωinjt+θ)iinj(t)dt\dfrac{d\theta}{dt}=\omega_0-\dfrac{M}{N}\omega_{inj}+\dfrac{1}{NT_{inj}}\displaystyle\int_{NT_{inj}}\!\!\tilde\Gamma\Big(\dfrac{M}{N}\omega_{inj}t+\theta\Big)i_{inj}(t)dt（Eq.29）；Ω(θ)=12Iinj∣Γ~N∣cos⁡(Nθ+∠Γ~N)\Omega(\theta)=\tfrac12 I_{inj}\vert\tilde\Gamma_N\vert\cos(N\theta+\angle\tilde\Gamma_N)（Eq.30）paper_004_injection_locking_part2paper_004 (Hong Part II 2019), Sec. IV, Eq.(28)–(30), p.2129（已核實）「M:N 次諧波鎖定與 ILFD」節final (verified)
52[P4] p.2130：M:N lock range（÷N ILFD）ωL=12Iinj∣Γ~N∣=Iinj cN2qmax\omega_L=\dfrac12 I_{inj}\vert\tilde\Gamma_N\vert=\dfrac{I_{inj}\,c_N}{2q_{max}}（半波對稱 ⇒c2=0⇒\Rightarrow c_2=0\Rightarrow 不能 ÷2）paper_004_injection_locking_part2paper_004, p.2130（原文逐字："which can be calculated from (30) to be ωL=Iinj∣Γ~N∣/2\omega_L=I_{inj}\vert\tilde\Gamma_N\vert/2"，已核實）「M:N 次諧波鎖定與 ILFD」節 第 4 步final (verified)
53ring 差動：功率/頻率/雜訊記帳（[P2] Eq.31–33）P=NItailVDDP=N I_{tail}V_{DD}（Eq.31）；f0≈Itail2ηNqmaxf_0\approx\dfrac{I_{tail}}{2\eta N q_{max}}（Eq.32）；in2‾Δf=4kTItail(1Vchar+1RLItail)\dfrac{\overline{i_n^2}}{\Delta f}=4kTI_{tail}\Big(\dfrac{1}{V_{char}}+\dfrac{1}{R_LI_{tail}}\Big)（Eq.33）lc_vs_ringpaper_002, Sec. V-B, Eq.(31)–(33), p.796（逐字核實）第 2b 步 (a)(b)(d)final (verified)
54ring 差動：phase noise/jitter 下限（明含 NN）（[P2] Eq.34–35）Lmin{Δf}=83η⋅N⋅kTP(VDDVchar+VDDRLItail)f02Δf2\mathcal{L}_{min}\{\Delta f\}=\dfrac{8}{3\eta}\cdot N\cdot\dfrac{kT}{P}\Big(\dfrac{V_{DD}}{V_{char}}+\dfrac{V_{DD}}{R_LI_{tail}}\Big)\dfrac{f_0^2}{\Delta f^2}（Eq.34）；κmin∝N\kappa_{min}\propto\sqrt{N}（Eq.35）lc_vs_ringpaper_002, Sec. V-B, Eq.(34)–(35), p.796（逐字核實）；固定 P,f0P,f_0 下與 single-ended 的 NN-independence（Eq.23）相反，phase noise 隨 NN 惡化第 2b 步 (e)final (verified)
55ring 相關供電/基板雜訊選擇律（[P2] Eq.37–38）ΓΣ(x)≡∑n=0N−1Γ(x+2πnN)\Gamma_\Sigma(x)\equiv\displaystyle\sum_{n=0}^{N-1}\Gamma\Big(x+\dfrac{2\pi n}{N}\Big)（Eq.37）；cΣ,m=Ncmc_{\Sigma,m}=Nc_m（m≡0 mod Nm\equiv0\bmod N）或 00（否則）（Eq.38）lab_34_correlated_supplypaper_002, Sec. VI, Eq.(37)–(38), p.797（逐字轉錄；本站補齊有限幾何級數證明）第 2.1–2.2 節final (verified)
56ADEV flicker-FM floor（本站自推前因子）σy2=2ln⁡2⋅h−1 (τ-無關)σy,floor=2ln⁡2⋅h−1≈1.1774h−1\sigma_y^2=2\ln2\cdot h_{-1}\ (\tau\text{-無關})\quad\sigma_{y,\text{floor}}=\sqrt{2\ln2\cdot h_{-1}}\approx1.1774\sqrt{h_{-1}}allan_variance本站從 ADEV 頻域積分核 I3=∫0∞sin⁡4u/u3 du=ln⁡2I_3=\int_0^\infty\sin^4u/u^3\,du=\ln2 自行推導；與 IEEE Std 1139 / NIST SP 1065 標準結果一致（外部文獻）「完整前因子表」節final (verified)
57PLL type-II peaking 閉式解∣Hlp∣max2=(s+1)2(s−1)(s+3), fpk=fn2s+1, s=1+8ζ2\lvert H_{lp}\rvert^2_{max}=\dfrac{(s+1)^2}{(s-1)(s+3)},\ f_{pk}=f_n\sqrt{\dfrac{2}{s+1}},\ s=\sqrt{1+8\zeta^2}（ζ=1/2\zeta=1/\sqrt2 時峰值恰為黃金比例 φ=1.618\varphi=1.618，即 2.092.09 dB）pll_noise_budget本站對規範 10.2 的 ∣Hlp∣2\vert H_{lp}\rvert^2 自行代數求極值（自含推導）；ζ↔\zeta\leftrightarrow phase margin 對照與級聯 0.1 dB 法則屬標準控制/電信文獻（外部）「補充推導：peaking 的閉式解」節final (verified)
58FOM 理論天花板FOM=173.8 dB−10log⁡10Feff(T=300 K，此為 1⋅kT 非 2kT)\mathrm{FOM}=173.8\ \text{dB}-10\log_{10}F_{eff}\quad(T=300\text{ K，此為 }1\cdot kT\text{ 非 }2kT)fom_limit本站從 paper_001 Eq.(21) 與 paper_002 Eq.(23) 的萬用形 Llin=Feff kT/P (f0/Δf)2\mathcal{L}_{lin}=F_{eff}\,kT/P\,(f_0/\Delta f)^2 推導；Cref=173.83C_{ref}=173.83 dB 為 k=1.380649×10−23k=1.380649\times10^{-23} J/K、T=300T=300 K 之直接計算第 0–1 步final (verified)
59SNR from aperture jitter（ADC，外部標準結果）SNRjitter=−20log⁡10(2πfinσt) [dB]\mathrm{SNR}_{jitter}=-20\log_{10}(2\pi f_{in}\sigma_t)\ [\text{dB}]；ENOB=(SNR−1.76)/6.02\mathrm{ENOB}=(\mathrm{SNR}-1.76)/6.02adc_aperture_jitter資料轉換器標準教科書結果（外部文獻，非本站 5 篇 PDF）；本站接回 L(Δf)→σt→SNR\mathcal{L}(\Delta f)\to\sigma_t\to\mathrm{SNR} 主線第 3 步「SNR：招牌公式」reference
60TJ(BER) dual-Dirac 外插公式TJ(BER)=DJδδ+2 Q−1(BER) σ\mathrm{TJ}(\mathrm{BER})=\mathrm{DJ}_{\delta\delta}+2\,Q^{-1}(\mathrm{BER})\,\sigmadj_dual_diracSerDes 業界標準模型（外部文獻，非本站 5 篇 PDF）；本站逐步從 Q 函數尾巴積分推導第 6 步「TJ(BER) 外插公式」reference
61×N / ÷N 相位縮放規則理想 ×N\times N：Lout=Lin+20log⁡10N\mathcal{L}_{out}=\mathcal{L}_{in}+20\log_{10}N；理想 ÷N\div N：Lout=Lin−20log⁡10N\mathcal{L}_{out}=\mathcal{L}_{in}-20\log_{10}Nclock_chain_budget標準頻率合成結果（外部文獻）；本站逐步從 ϕout=Nϕin\phi_{out}=N\phi_{in} 或 ϕin/N\phi_{in}/N 與 Sϕ∝ϕ2S_\phi\propto\phi^2 推導「規則 1」「規則 2」節final
62注入鎖定 OU 雜訊整形（一階 PLL）與鎖外拍頻Sθ(ω)=Snωc2+ω2S_\theta(\omega)=\dfrac{S_n}{\omega_c^2+\omega^2}（corner ωc=ωL2−Δω2\omega_c=\sqrt{\omega_L^2-\Delta\omega^2}）；鎖外 ωb=Δω2−ωL2\omega_b=\sqrt{\Delta\omega^2-\omega_L^2}（[P4] Eq.34）injection_locking_noisecorner 為 [P3] Eq.(40) 之 pull-in frequency 的原生結果；把雜訊掛上 Adler 讀出整形 PSD 為標準注入鎖定雜訊理論（外部文獻：Kurokawa 1973，非本站 5 篇 PDF）；ωb\omega_b 為本站從 Adler 方程分離變數逐步積分導出，與 [P4] Eq.(34), p.2130 一致Part A 第 3 步；Part B 第 2 步final (verified)

v9 新增公式（次諧波／大注入倍頻器）​

本節整合次諧波注入（ILCM，injection-locked clock multiplier）與 [P4] 大注入剩料波次新增的 headline 公式；LaTeX 逐字取自對應教學頁，不重新推導。

#ConceptFinal Formula推導頁 Derivation來源 Source站內出處Notes
63次諧波（×N）倍頻 lock range——由 [P4] Eq.(29) 推出的閉式Ω(θ)≈12∣IN∣∣Γ~1∣cos⁡(θ+∠Γ~1−∠IN),ωL=12∣IN∣ ∣Γ~1∣\Omega(\theta)\approx\frac12\vert I_N\vert\vert\tilde\Gamma_1\vert\cos\big(\theta+\angle\tilde\Gamma_1-\angle I_N\big),\qquad\omega_L=\frac12\vert I_N\vert\,\vert\tilde\Gamma_1\vertsubharmonic_injection本站從 [P4] Eq.(29), p.2129（已核實）代入 (M,N)[P4]=(N,1)(M,N)_{[P4]}=(N,1) 逐項平均推出；與 [P3] Sec. IV footnote 7（p.2112，已核實）的脈衝列算術逐項對帳「路線一」第 3 步final (verified)
64realignment factor β\beta（一根脈衝拉回的比例）β≡−qinj Γ~′(θss)\beta\equiv-q_{inj}\,\tilde\Gamma'(\theta_{ss})（穩定 0<β<20\lt\beta\lt2；β/Tinj=ωc=−Ω′(θss)\beta/T_{inj}=\omega_c=-\Omega'(\theta_{ss})＝[P3] Eq.(40) 的 pull-in frequency）subharmonic_injection本站從線性化 per-pulse map θk+1=θk+Δω0NT0+qinjΓ~(θk)\theta_{k+1}=\theta_k+\Delta\omega_0NT_0+q_{inj}\tilde\Gamma(\theta_k) 推出（[P3] Sec. IV footnote 7 的離散算術延伸）第 3 節final (verified)
65一階離散時間迴路的雜訊 corner（Δω=0\Delta\omega=0 時 =ΔωL=\Delta\omega_L）fc=β1−β⋅fref2π≈βfref2πf_c=\dfrac{\beta}{1-\beta}\cdot\dfrac{f_{ref}}{2\pi}\approx\dfrac{\beta f_{ref}}{2\pi}；精確離散閉式 fc′=fref2πarccos⁡ ⁣(1−β22(1+β))f_c'=\dfrac{f_{ref}}{2\pi}\arccos\!\big(1-\dfrac{\beta^2}{2(1+\beta)}\big)subharmonic_injection本站從 Hosc(z)=(1−z−1)/(1−(1−β)z−1)H_{osc}(z)=(1-z^{-1})/(1-(1-\beta)z^{-1}) 的 ∣H∣2=1/2\vert H\vert^2=1/2 點求出；與 injection_locking_noise 的連續版 ωc=ωL2−Δω2\omega_c=\sqrt{\omega_L^2-\Delta\omega^2} 在 f≪freff\ll f_{ref} 一致第 4.2 節final (verified)
66輸出 jitter 閉式（時間平均相位方差）σout2=σw2[(1−β)2β(2−β)+12]=κ2NT0⋅1−β+β2/2β(2−β)\sigma_{out}^2=\sigma_w^2\Big[\dfrac{(1-\beta)^2}{\beta(2-\beta)}+\dfrac12\Big]=\kappa^2NT_0\cdot\dfrac{1-\beta+\beta^2/2}{\beta(2-\beta)}（σw2=κ2Tinj\sigma_w^2=\kappa^2T_{inj}；MC 比值 0.999）subharmonic_injection本站從線性化 map 的幾何級數 θk−=∑j(1−β)jwk−j\theta_k^-=\sum_j(1-\beta)^jw_{k-j} 逐步推出；κ2\kappa^2 為 [P2] Eq.(11), p.793（已核實）的方差成長率第 4.3 節final (verified)
67Generalized Adler（Mirzaei）與大注入 lock range（[P4] Eq.(8)–(9)）dθdt=ω0−ωinj−ω02QIinjsin⁡θIosc+Iinjcos⁡θ\dfrac{d\theta}{dt}=\omega_0-\omega_{inj}-\dfrac{\omega_0}{2Q}\dfrac{I_{inj}\sin\theta}{I_{osc}+I_{inj}\cos\theta}（Eq.8）；ωL=ω02QIinjIosc11−Iinj2/Iosc2\omega_L=\dfrac{\omega_0}{2Q}\dfrac{I_{inj}}{I_{osc}}\dfrac{1}{\sqrt{1-I_{inj}^2/I_{osc}^2}}（Eq.9）paper_004_large_injection_transientpaper_004 (Hong Part II 2019), Sec. III-A, p.2123（逐字轉錄，已核實）第 1.1 節final (verified)
68反比於振幅的有效 ISF（[P4] Eq.(13)）Γ~LC=Γ~1+A\tilde\Gamma_{LC}=\dfrac{\tilde\Gamma}{1+A}paper_004_large_injection_transientpaper_004, Sec. III-C, p.2124（逐字轉錄，已核實）第 1.2 節final (verified)

圖例​

  • final：該主題的最終結果公式。
  • step：推導過程中的中間步驟。
  • reference：作為對照/比較的外部模型（未必出自下載的 5 篇 PDF）。
  • ⚠️：manual_verification_needed = true，請對照原始 PDF 再確認。