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

Equation Index

This page is auto-generated by scripts/build_equation_index.py from extracted/extracted_equations.json. Entries flagged ⚠️ mean the exact constants/forms still need manual comparison against the PDF.

#	Concept	Final Formula	Derivation	Source	Notes
1	impulse charge	$\Delta q = \int i(t)\, dt$	impulse_to_phase_shift	paper_001 (general.pdf), around Eq. (9), p.182	step
2	charge to voltage step	$\Delta V = \dfrac{\Delta q}{C_{node}}$	impulse_to_phase_shift	paper_001, Eq. (9), p.182	step
3	impulse-to-phase (ISF definition in operational form)	$\Delta\phi = \dfrac{\Gamma(\omega_0\tau)}{q_{max}}\,\Delta q$	impulse_to_phase_shift	paper_001, from Eq. (10)-(11), p.182	final
4	excess-phase impulse response (LTV)	$h_\phi(t,\tau) = \dfrac{\Gamma(\omega_0\tau)}{q_{max}}\,u(t-\tau)$	isf_definition	paper_001, Eq. (10), p.182	final
5	LTV phase response (convolution)	$\phi(t) = \dfrac{1}{q_{max}}\int_{-\infty}^{t}\Gamma(\omega_0\tau)\,i_n(\tau)\,d\tau$	convolution_derivation	paper_001, Eq. (11), p.182	final
6	ISF Fourier series	$\Gamma(\omega_0\tau) = \dfrac{c_0}{2} + \sum_{n=1}^{\infty} c_n\cos(n\omega_0\tau+\theta_n)$	fourier_series_of_isf	paper_001, Eq. (12), p.183	final
7	rms ISF (Parseval)	$\sum_{n=0}^{\infty} c_n^2 = \dfrac{1}{\pi}\int_0^{2\pi}\vert \Gamma(x)\vert ^2 dx = 2\,\Gamma_{rms}^2$	rms_isf	paper_001, Eq. (20), p.185	final
8	white noise phase noise (1/f^2)	$\mathcal{L}\{\Delta\omega\} = 10\log_{10}\!\left(\dfrac{\Gamma_{rms}^2}{q_{max}^2}\cdot\dfrac{\overline{i_n^2}/\Delta f}{4\,\Delta\omega^2}\right)$	white_noise_to_phase_noise	paper_001, Eq. (21), p.185	final
9	flicker noise upconversion (1/f^3)	$\mathcal{L}\{\Delta\omega\} = 10\log_{10}\!\left(\dfrac{c_0^2}{q_{max}^2}\cdot\dfrac{\overline{i_n^2}/\Delta f}{8\,\Delta\omega^2}\cdot\dfrac{\omega_{1/f}}{\Delta\omega}\right)$	flicker_noise_upconversion	paper_001, Eq. (23), p.185	final
10	1/f^3 corner	$\Delta\omega_{1/f^3} = \omega_{1/f}\cdot\dfrac{c_0^2}{2\,\Gamma_{rms}^2}$	flicker_noise_upconversion	paper_001, Eq. (24), p.185	final
11	SSB phase noise vs phase PSD	$\mathcal{L}(\Delta f) \approx \tfrac{1}{2} S_\phi(\Delta f)$	psd_phase_noise_jitter	standard small-angle relation; consistent with paper_001 usage	final
12	phase error to timing error	$\Delta t = \dfrac{\Delta\phi}{2\pi f_0}$	psd_phase_noise_jitter	standard; used throughout paper_002 and SerDes practice	final
13	phase variance from PSD	$\sigma_\phi^2 = \int_{f_1}^{f_2} S_\phi(f)\, df$	psd_phase_noise_jitter	standard	final
14	rms jitter from phase variance	$\sigma_t = \dfrac{\sigma_\phi}{2\pi f_0} = \dfrac{1}{2\pi f_0}\sqrt{\int_{f_1}^{f_2} S_\phi(f)\, df}$	serdes_clocking_connection	standard; SerDes clocking	final
15	accumulated (ring) jitter random walk	$\sigma_{\Delta t} = \kappa\,\sqrt{\Delta t}$	lab_03_ring_oscillator_toy_model	paper_002, Eq. (8), p.792 (kappa via Eq.(12), p.793)	final
16	ring frequency vs stages	$f_0 = \dfrac{1}{2 N \tau_D}$	lc_vs_ring	paper_002, Eq. (14), p.794	final
17	ring rms ISF scaling	$\Gamma_{rms}=\sqrt{\dfrac{2\pi^2}{3\eta^3}}\;\dfrac{1}{N^{1.5}}\;\Rightarrow\;\Gamma_{rms}\propto N^{-3/2}\ (\Gamma_{rms}^2\propto N^{-3})$ (at $\eta=0.75$, $\approx 4/N^{1.5}$, i.e. the solid line in [P2] Fig.8; the radical covers only the constant)	lc_vs_ring	[P2] Eq.(16), p.794 (v7 re-verified: the radical covers only the constant, $\Gamma_{rms}\propto N^{-3/2}$; confirmed by triple evidence — the text's $4/N^{1.5}$@$\eta=0.75$ and App.B Eq.(55). v3 had misread it as $N^{-3/4}$)	final (verified)
18	ring phase noise FOM (white)	$\mathcal{L}\{\Delta f\}=\dfrac{8}{3\eta}\cdot\dfrac{kT}{P}\cdot\dfrac{V_{DD}}{V_{char}}\cdot\left(\dfrac{f_0}{\Delta f}\right)^2\quad(\text{min at }V_T=0:\ \tfrac{16\gamma}{3\eta})$	lc_vs_ring	paper_002, Eq. (23),(25), p.796. The prefactor is $8/(3\eta)$ ($\eta$ is the stage-delay proportionality constant of Eq.14, $\approx 1$); $\gamma$ enters only through $V_{char}=\Delta V/\gamma$. (v2 had erroneously changed it to $8/(3\gamma)$ and mislabeled it "verified verbatim"; v3 corrected it against the original PDF p.796.)	final (verified)
19	Leeson model (comparison only)	$\mathcal{L}(\Delta\omega) = 10\log_{10}\!\left[\dfrac{2FkT}{P_s}\left(1+\left(\dfrac{\omega_0}{2Q\Delta\omega}\right)^2\right)\left(1+\dfrac{\omega_{1/f^3}}{\vert \Delta\omega\vert }\right)\right]$	equation_index	Leeson (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
20	generalized Adler / phase equation (injection)	$\frac{d\theta}{dt}=(\omega_0-\omega_{inj})+\Omega(\theta),\quad \Omega(\theta)=\frac{1}{T_{inj}}\int_{T_{inj}}\tilde\Gamma(\omega_{inj} t+\theta)\,i_{inj}(t)\,dt,\ \ \tilde\Gamma=\Gamma/q_{max}$	paper_003_injection_locking_part1	paper_003 (Hong Part I 2019): $\tilde\Gamma=\Gamma/q_{max}$ Eq.(26); time-averaged generalized Adler equation Eq.(30), p.2113 (the original's averaged term carries a plus sign, averaging period $T_{inj}$); lock range $\omega_L=\tfrac{1}{2} I_{inj}\vert\tilde\Gamma_1\vert$ Eq.(35), p.2114. On this site $\Omega(\theta)$ adds with the same sign as the difference frequency $(\omega_0-\omega_{inj})$, matching the sign convention of [P3].	final (verified)
21	amplitude perturbation function (APF)	$\tilde\Lambda\ (\text{APF}):\ d(t,\phi)=e^{-t/\tau_0},\ \tau_0=\frac{2Q}{\omega_{osc}};\quad \tilde\Gamma_1=\tfrac{1}{q_{max}}\angle90^\circ,\ \tilde\Lambda_1=\tfrac{\tau_0}{q_{max}}\angle0^\circ\ (\text{quadrature})$	paper_004_injection_locking_part2	paper_004 (Hong Part II 2019), Eq.(25),(26),(27), p.2128 (verified verbatim)	final (verified)
22	PPV / adjoint (broader literature, NOT in the 5 PDFs)	$\dot{\phi}(t) = v_1^T(t)\, B(t)\, \xi(t)\quad\text{(Demir et al. PPV form)}$	effective_isf	Demir-Mehrotra-Roychowdhury (2000), IEEE TCAS-I 47(5):655-674, DOI 10.1109/81.847872 (citation verified; external, not in the 5 PDFs).	reference
23	ring kappa (jitter rate)	$\kappa=\dfrac{\Gamma_{rms}}{q_{max}}\sqrt{\dfrac{1}{2}\dfrac{\overline{i_n^2}}{\Delta f}}\quad(\sigma_{\Delta t}=\kappa\sqrt{\Delta t})$	lab_03_ring_oscillator_toy_model	paper_002, Eq. (12), p.793 (verified verbatim)	final (verified)
24	ring per-stage device noise	$\dfrac{\overline{i_n^2}}{\Delta f}=4kT\gamma\,g_{d0}=4kT\gamma\,\mu C_{ox}\dfrac{W}{L}\,\Delta V$	device_noise_mapping	paper_002, Eq. (17),(18), p.795 (verified verbatim)	final (verified)
25	Lorentzian carrier lineshape	$S(\Delta\omega)\propto\dfrac{D}{D^2+\Delta\omega^2},\quad \Delta f_{3\mathrm{dB}}=\dfrac{D}{\pi},\quad D=\dfrac{\Gamma_{rms}^2}{4q_{max}^2}\dfrac{\overline{i_n^2}}{\Delta f}=\dfrac{\kappa^2}{2}$	lorentzian_linewidth	phase 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)
26	Allan variance from PSD	$\sigma_y^2(\tau)=2\int_0^\infty S_y(f)\dfrac{\sin^4(\pi f\tau)}{(\pi f\tau)^2}\,df,\quad S_y=\dfrac{f^2}{f_0^2}S_\phi$	allan_variance	white/flicker/RW FM -> sigma_y ~ tau^-0.5, tau^0, tau^0.5. External: Allan 1966 [E1-ext].	final
27	PLL output phase-noise budget	$S_{out}=(S_{ref}N^2+S_{cp})\lvert H_{lp}\rvert^2+S_{vco}\lvert H_{hp}\rvert^2$	pll_noise_budget	sum of all PLL noise sources shaped by their transfer; optimal loop BW minimizes integrated jitter.	final
28	ISF as harmonic transfer vector (HTM)	$\text{input at } f \to \text{output at } f+kf_0 \text{ with gain } c_k\ (\text{ISF Fourier coeff})$	ltv_htm	rigorous LTV / harmonic transfer matrix view; ISF is the phase output's HTM row. External: Zadeh 1950 [E5].	reference

v5–v8 additions (jitter kernels, diffusion dictionary, asymmetric closed form, [P3]/[P4] advanced, ring correlated noise)

This section consolidates the headline equations added in the v5–v8 development waves. Columns match the main table above, plus an "On-site source" column naming the derivation section within the teaching page. Each row is transcribed verbatim from the corresponding teaching page (not re-derived here); the teaching pages themselves have already been checked against the paper PDFs.

#	Concept	Final Formula	Derivation	Source	On-site source	Notes
29	timing error = sampled phase (TIE kernel)	$\sigma_{\text{TIE}}^2=\dfrac{1}{\omega_0^2}\displaystyle\int_{f_1}^{f_2}S_\phi(f)\,df$	jitter_kernels	derived on this site; isomorphic to paper_002 Eq.(46)-(49), p.803 (Khinchin route)	Step 3, "Kernel (a): TIE"	final
30	N-period jitter kernel $4\sin^2(\pi fNT)$	$\sigma_P^2(N)=\dfrac{1}{\omega_0^2}\displaystyle\int_0^\infty S_\phi(f)\,4\sin^2(\pi fNT)\,df$	jitter_kernels	derived on this site; the kernel is equivalent (verified verbatim via the one-sided/two-sided conversion) to paper_002 Eq.(49), p.803 (one-sided $4\sin^2$)	Step 3, "Kernel (b): N-period"	final (verified)
31	cycle-to-cycle jitter kernel $16\sin^4(\pi fT)$	$\sigma_{c2c}^2=\dfrac{1}{\omega_0^2}\displaystyle\int_0^\infty S_\phi(f)\,16\sin^4(\pi fT)\,df$	jitter_kernels	derived on this site; paper_002 Eq.(51), p.803 (cycle-to-cycle; this site distinguishes two competing definitions)	Step 3, "Kernel (c): cycle-to-cycle"	final (verified)
32	white-FM closed form (kernel integral = [P2] random walk)	$\sigma_{\Delta\phi}^2(N)=\kappa^2\,NT\quad(\kappa^2=\Gamma_{rms}^2 S_i/(2q_{max}^2))$	jitter_kernels	paper_002, Eq.(8)/(11)/(12), p.792-793 (verified); this site's frequency-domain kernel integral recovers it exactly	Step 4, "White-FM closed form"	final (verified)
33	[P2] Eq.(45)-(51), p.803 (jitter from phase spectrum, autocorrelation/Khinchin)	$R_\phi(\tau)=\displaystyle\int_{-\infty}^{\infty}S_\phi(f)e^{j2\pi f\tau}df$ (two-sided, Eq.48); $\sigma_{\Delta\phi}^2=\dfrac{8}{\omega_0^2}\displaystyle\int_0^\infty S_\phi(f)\sin^2(\pi f\tau)\,df$ (Eq.49); $\kappa=\dfrac{\Delta f}{f_0}10^{-\mathcal{L}\{\Delta f\}/20}$ (Eq.50)	jitter_kernels	paper_002, Eq.(45)-(51), p.803 (v5 verified verbatim; see section "Which papers/equations this corresponds to")	Section "Which papers/equations this corresponds to"	final (verified)
34	diffusion dictionary's protagonist $\kappa^2$ ([P2] Eq.11)	$\kappa^2\equiv\dfrac{\Gamma_{rms}^2}{2q_{max}^2}\cdot\dfrac{\overline{i_n^2}}{\Delta f}\ [\text{rad}^2/\text{s}]$ (canonical $0.125$)	diffusion_dictionary	paper_002, Eq.(11), p.793 (verified verbatim)	Step 0, "There is only one protagonist"	final (verified)
35	$D=\kappa^2/2$ (convention B, adopted by this site's spec)	$D=\dfrac{\Gamma_{rms}^2}{4q_{max}^2}\dfrac{\overline{i_n^2}}{\Delta f}=\dfrac{\kappa^2}{2}$ (canonical $0.0625$)	diffusion_dictionary	corresponds to the Demir 2000 convention (external literature); v5 correction (v3 had mistakenly used convention A's value $D=\kappa^2$)	Section "Outfit two: D"	final (verified)
36	Lorentzian FWHM $=\kappa^2/2\pi$ (=D/π, v5 mapping)	$\Delta f_{3\mathrm{dB}}=\dfrac{\kappa^2}{2\pi}=\dfrac{D}{\pi}$ (canonical $19.9$ mHz; true LC $39.8$ mHz)	diffusion_dictionary	derived from paper_002 Eq.(11)'s $\kappa^2$; external literature Demir 2000 [E2] for the Lorentzian mechanism	Section "Outfit three: Lorentzian 3-dB linewidth"	final (verified)
37	$b_{-2}$: 1/f² phase PSD coefficient	$S_\phi(f)=\dfrac{2\kappa^2}{(2\pi f)^2}\equiv\dfrac{b_{-2}}{f^2}\ [\text{rad}^2/\text{Hz}], \ b_{-2}=\dfrac{\kappa^2}{2\pi^2}$ (canonical $6.33\times10^{-3}$)	diffusion_dictionary	consistent with paper_001 Eq.(21), p.185 (the $/4$ version) and the time-domain derivation in white_noise_to_phase_noise	Section "Outfit four: 1/f² phase PSD coefficient"	final (verified)
38	App.B $\Gamma_{rms}^2(N,A)$ closed form (asymmetric triangular ISF)	$\Gamma_{rms}^2=\dfrac{2\pi^2}{3\eta^3}\dfrac{1}{N^3}\Big[4\dfrac{1+A^3}{(1+A)^3}\Big]$ (exactly reduces to Eq.16 at $A=1$)	asymmetric_isf_closed_form	paper_002, App. B, Eq.(52)-(55), p.803 (verified verbatim against this render pass, v8)	Steps 1-3	final (verified)
39	App.B $\Gamma_{dc}(N,A)$ closed form (DC value / source of $c_0$)	$\Gamma_{dc}=\dfrac{2\pi}{\eta^2}\dfrac{1}{N^2}\Big(\dfrac{1-A}{1+A}\Big)\quad(c_0=2\Gamma_{dc})$	asymmetric_isf_closed_form	paper_002, App. B, Eq.(56), p.803 (verified verbatim against this render pass, v8)	Step 4	final (verified)
40	App.B 1/f³ corner closed form (computed directly from $N,A$)	$f_{1/f^3}=f_{1/f}\cdot\dfrac{3}{2\eta N}\cdot\dfrac{(1-A)^2}{1-A+A^2}$ ([P1] Eq.24 convention value $=2\times$ this formula)	asymmetric_isf_closed_form	paper_002, App. B, Eq.(57), p.803 (verified verbatim against this render pass, v8; corresponds to main-text Eq.7, p.792)	Step 5	final (verified)
41	[P1] Appendix Eq.(31), tangential projection	$l=\Delta\vec X\cdot\dfrac{\dot{\vec X}}{\lvert\dot{\vec X}\rvert}$	isf_from_waveform	paper_001, Appendix, Eq.(31), p.192 (verbatim transcription)	Method B, Step 1	final (verified)
42	[P1] Appendix Eq.(32)-(33): displacement -> time -> phase	$\Delta t=\dfrac{l}{\lvert\dot{\vec X}\rvert}=\Delta\vec X\cdot\dfrac{\dot{\vec X}}{\lvert\dot{\vec X}\rvert^2}$ (Eq.32); $\Delta\phi=\dfrac{2\pi}{T}\Delta t$ (Eq.33)	isf_from_waveform	paper_001, Appendix, Eq.(32)-(33), p.193 (verbatim transcription)	Method B, Steps 2-3	final (verified)
43	[P1] Appendix Eq.(34)-(36): node-voltage special case, closed-form ISF	$\Delta\phi_i=\dfrac{2\pi}{T}\dfrac{\Delta q_i}{C_i}\dfrac{\dot v_i}{\lvert\dot{\vec v}\rvert^2}$ (Eq.34); $\Gamma_i(x)=\dfrac{f_i'}{\sum_j f_j'^{\,2}}$ (Eq.36)	isf_from_waveform	paper_001, Appendix, Eq.(34)-(36), p.193 (verbatim transcription)	Method B, Steps 4-5	final (verified)
44	[P1] Appendix Eq.(37): second-order-system closed form ($f=\cos x\Rightarrow\Gamma=-\sin x$)	$\Gamma(x)=\dfrac{f'}{f'^{\,2}+f''^{\,2}}$	isf_from_waveform	paper_001, Appendix, Eq.(37), p.193 (verbatim transcription; verified via the sinusoid self-check)	Method B, Step 6	final (verified)
45	[P1] Appendix Eq.(38): first-derivative approximation (ring-specific)	$\Gamma_i(x)=\dfrac{f_i'(x)}{f_{max}'^{\,2}}$	isf_from_waveform	paper_001, Appendix, Eq.(38), p.193 (verbatim transcription)	Method C	final (verified)
46	[P3] Sec. IV impulse-train Eq.(19)-(21): kick and frequency shift	$\Delta\phi=\pm\dfrac{q_{inj}}{q_{max}}$ (Eq.19); $\Delta\omega=\dfrac{\Delta\phi}{T_{inj}}$ (Eq.20-21)	paper_003_injection_locking_part1	paper_003 (Hong Part I 2019), Sec. IV, Eq.(19)-(21), p.2112 (verified)	Section "Locking to an impulse train", Steps 1-2	final (verified)
47	[P3] Eq.(22)-(23): energy balance and impulse-train fundamental amplitude	$\omega_0 q_{max}=Q\,I_{osc}$ (Eq.22); $I_{inj}=\dfrac{2q_{inj}}{T_{inj}}$ (Eq.23)	paper_003_injection_locking_part1	paper_003, Sec. IV, Eq.(22)-(23), p.2112 (verified)	Section "Locking to an impulse train", Step 4	final (verified)
48	Lab 36 / [P3] Eq.(38)-(40): acquisition transient, pull-in frequency	$\omega_c\equiv\omega_L\cos\theta_{ss}=\sqrt{\omega_L^2-\Delta\omega^2}$ (the $\omega_p$ of [P3] Eq.40); exact solution $R(\theta(t))=R(\theta_0)e^{-\omega_c t}$	lab_36_lock_acquisition	paper_003, Eq.(38)-(40), p.2115 (verified); this site's exact closed-form solution is derived on-site and is equivalent to the tanh form of [P4] Eq.(31)-(32)	Section 2.1, "Part (a) acquisition transient"	final (verified)
49	Lab 36: tilted-washboard barrier and Kramers slip rate (external literature)	$\Delta U_+=2\omega_L\big[\sqrt{1-r^2}-r\arccos r\big]$; $\nu_{slip}\approx\dfrac{\omega_c}{2\pi}e^{-\Delta U_+/D}$	lab_36_lock_acquisition	the barrier is derived on this site (from $U(\theta)=-\Delta\omega\theta-\omega_L\cos\theta$); the escape rate is Kramers 1940, Risken 1989 (external literature, not in the 5 PDFs on this site)	Section 2.2, "Part (b) tilted washboard"	reference
50	[P3] Eq.(43)-(45): optimal injection waveform and lock-range bound (Cauchy-Schwarz)	$I_{rms}\equiv\sqrt{\langle i_{inj}^2\rangle}$ (Eq.43); $i_{inj,0}^{*}(x)=\pm\dfrac{I_{rms}}{\tilde\Gamma_{rms}}\tilde\Gamma(x)$ (Eq.44); $\omega_L^*=I_{rms}\tilde\Gamma_{rms}$ (Eq.45)	injection_locking_noise	paper_003, Sec. VI, Eq.(43)-(45), pp.2119-2120 (verified verbatim)	Section "Injection waveform design", Steps 0-3	final (verified)
51	[P4] Eq.(28)-(30): M:N subharmonic-locking lock characteristic	$\varphi(t)\equiv\dfrac{M}{N}\omega_{inj}t+\theta(t)$ (Eq.28); $\dfrac{d\theta}{dt}=\omega_0-\dfrac{M}{N}\omega_{inj}+\dfrac{1}{NT_{inj}}\displaystyle\int_{NT_{inj}}\!\!\tilde\Gamma\Big(\dfrac{M}{N}\omega_{inj}t+\theta\Big)i_{inj}(t)dt$ (Eq.29); $\Omega(\theta)=\tfrac12 I_{inj}\vert\tilde\Gamma_N\vert\cos(N\theta+\angle\tilde\Gamma_N)$ (Eq.30)	paper_004_injection_locking_part2	paper_004 (Hong Part II 2019), Sec. IV, Eq.(28)-(30), p.2129 (verified)	Section "M:N subharmonic locking and ILFD"	final (verified)
52	[P4] p.2130: M:N lock range (÷N ILFD)	$\omega_L=\dfrac12 I_{inj}\vert\tilde\Gamma_N\vert=\dfrac{I_{inj}\,c_N}{2q_{max}}$ (half-wave symmetry $\Rightarrow c_2=0\Rightarrow$ cannot divide by 2)	paper_004_injection_locking_part2	paper_004, p.2130 (verbatim: "which can be calculated from (30) to be $\omega_L=I_{inj}\vert\tilde\Gamma_N\vert/2$", verified)	Section "M:N subharmonic locking and ILFD", Step 4	final (verified)
53	Differential ring: power/frequency/noise accounting ([P2] Eq.31-33)	$P=N I_{tail}V_{DD}$ (Eq.31); $f_0\approx\dfrac{I_{tail}}{2\eta N q_{max}}$ (Eq.32); $\dfrac{\overline{i_n^2}}{\Delta f}=4kTI_{tail}\Big(\dfrac{1}{V_{char}}+\dfrac{1}{R_LI_{tail}}\Big)$ (Eq.33)	lc_vs_ring	paper_002, Sec. V-B, Eq.(31)-(33), p.796 (verified verbatim)	Step 2b, (a)(b)(d)	final (verified)
54	Differential ring: phase-noise/jitter floor (explicitly contains $N$) ([P2] Eq.34-35)	$\mathcal{L}_{min}\{\Delta f\}=\dfrac{8}{3\eta}\cdot N\cdot\dfrac{kT}{P}\Big(\dfrac{V_{DD}}{V_{char}}+\dfrac{V_{DD}}{R_LI_{tail}}\Big)\dfrac{f_0^2}{\Delta f^2}$ (Eq.34); $\kappa_{min}\propto\sqrt{N}$ (Eq.35)	lc_vs_ring	paper_002, Sec. V-B, Eq.(34)-(35), p.796 (verified verbatim); at fixed $P,f_0$ this is the opposite of the single-ended $N$-independence (Eq.23) — phase noise degrades with $N$	Step 2b, (e)	final (verified)
55	Ring correlated supply/substrate noise selection rule ([P2] Eq.37-38)	$\Gamma_\Sigma(x)\equiv\displaystyle\sum_{n=0}^{N-1}\Gamma\Big(x+\dfrac{2\pi n}{N}\Big)$ (Eq.37); $c_{\Sigma,m}=Nc_m$ (for $m\equiv0\bmod N$) or $0$ (otherwise) (Eq.38)	lab_34_correlated_supply	paper_002, Sec. VI, Eq.(37)-(38), p.797 (verbatim transcription; this site supplies the missing finite-geometric-series proof)	Sections 2.1-2.2	final (verified)
56	ADEV flicker-FM floor (prefactor self-derived on this site)	$\sigma_y^2=2\ln2\cdot h_{-1}\ (\text{independent of }\tau)\quad\sigma_{y,\text{floor}}=\sqrt{2\ln2\cdot h_{-1}}\approx1.1774\sqrt{h_{-1}}$	allan_variance	derived on this site from the ADEV frequency-domain integral kernel $I_3=\int_0^\infty\sin^4u/u^3\,du=\ln2$; consistent with the standard result in IEEE Std 1139 / NIST SP 1065 (external literature)	Section "Complete prefactor table"	final (verified)
57	PLL type-II peaking closed form	$\lvert H_{lp}\rvert^2_{max}=\dfrac{(s+1)^2}{(s-1)(s+3)},\ f_{pk}=f_n\sqrt{\dfrac{2}{s+1}},\ s=\sqrt{1+8\zeta^2}$ (at $\zeta=1/\sqrt2$ the peak is exactly the golden ratio $\varphi=1.618$, i.e. $2.09$ dB)	pll_noise_budget	this site derives the extremum of $\vert H_{lp}\rvert^2$ from spec Section 10.2 algebraically (self-contained derivation); the $\zeta\leftrightarrow$ phase-margin correspondence and the cascaded-0.1-dB rule belong to standard control/telecom literature (external)	Section "Supplementary derivation: closed form for peaking"	final (verified)
58	FOM theoretical ceiling	$\mathrm{FOM}=173.8\ \text{dB}-10\log_{10}F_{eff}\quad(T=300\text{ K; this is }1\cdot kT\text{, not }2kT)$	fom_limit	derived on this site from the universal form $\mathcal{L}_{lin}=F_{eff}\,kT/P\,(f_0/\Delta f)^2$ implicit in paper_001 Eq.(21) and paper_002 Eq.(23); $C_{ref}=173.83$ dB is a direct calculation from $k=1.380649\times10^{-23}$ J/K, $T=300$ K	Steps 0-1	final (verified)
59	SNR from aperture jitter (ADC, external standard result)	$\mathrm{SNR}_{jitter}=-20\log_{10}(2\pi f_{in}\sigma_t)\ [\text{dB}]$; $\mathrm{ENOB}=(\mathrm{SNR}-1.76)/6.02$	adc_aperture_jitter	standard data-converter textbook result (external literature, not in the 5 PDFs on this site); this site connects it back to the $\mathcal{L}(\Delta f)\to\sigma_t\to\mathrm{SNR}$ main line	Step 3, "SNR: the headline formula"	reference
60	TJ(BER) dual-Dirac extrapolation formula	$\mathrm{TJ}(\mathrm{BER})=\mathrm{DJ}_{\delta\delta}+2\,Q^{-1}(\mathrm{BER})\,\sigma$	dj_dual_dirac	SerDes industry-standard model (external literature, not in the 5 PDFs on this site); this site derives it step by step from the Q-function tail integral	Step 6, "TJ(BER) extrapolation formula"	reference
61	×N / ÷N phase-scaling rules	ideal $\times N$: $\mathcal{L}_{out}=\mathcal{L}_{in}+20\log_{10}N$; ideal $\div N$: $\mathcal{L}_{out}=\mathcal{L}_{in}-20\log_{10}N$	clock_chain_budget	standard frequency-synthesis result (external literature); this site derives it step by step from $\phi_{out}=N\phi_{in}$ or $\phi_{in}/N$ together with $S_\phi\propto\phi^2$	Sections "Rule 1" and "Rule 2"	final
62	Injection-locked OU noise shaping (first-order PLL) and out-of-lock beat frequency	$S_\theta(\omega)=\dfrac{S_n}{\omega_c^2+\omega^2}$ (corner $\omega_c=\sqrt{\omega_L^2-\Delta\omega^2}$); out-of-lock $\omega_b=\sqrt{\Delta\omega^2-\omega_L^2}$ ([P4] Eq.34)	injection_locking_noise	the corner is a native consequence of [P3] Eq.(40)'s pull-in frequency; attaching noise to the Adler equation and reading off the shaped PSD is standard injection-locking noise theory (external literature: Kurokawa 1973, not in the 5 PDFs on this site); $\omega_b$ is derived on this site by separation of variables and step-by-step integration of the Adler equation, consistent with [P4] Eq.(34), p.2130	Part A, Step 3; Part B, Step 2	final (verified)

v9 additions (subharmonic / large-injection multiplier)

This section collects the headline formulas added by the subharmonic-injection (ILCM, injection-locked clock multiplier) and [P4] large-injection leftover waves; the LaTeX is copied verbatim from the corresponding teaching pages, not re-derived.

#	Concept	Final Formula	推導頁 Derivation	來源 Source	On-site location	Notes
63	Subharmonic (×N) multiplier lock range — closed form derived from [P4] Eq.(29)	$\Omega(\theta)\approx\frac12\vert I_N\vert\vert\tilde\Gamma_1\vert\cos\big(\theta+\angle\tilde\Gamma_1-\angle I_N\big),\qquad\omega_L=\frac12\vert I_N\vert\,\vert\tilde\Gamma_1\vert$	subharmonic_injection	derived on this site from [P4] Eq.(29), p.2129 (verified) by substituting $(M,N)_{[P4]}=(N,1)$ and averaging term by term; cross-checked against the impulse-train arithmetic of [P3] Sec. IV footnote 7 (p.2112, verified)	"Route 1," Step 3	final (verified)
64	Realignment factor $\beta$ (the fraction of phase error one pulse pulls back)	$\beta\equiv-q_{inj}\,\tilde\Gamma'(\theta_{ss})$ (stable for $0\lt\beta\lt2$; $\beta/T_{inj}=\omega_c=-\Omega'(\theta_{ss})$ = the pull-in frequency of [P3] Eq.(40))	subharmonic_injection	derived on this site from the linearized per-pulse map $\theta_{k+1}=\theta_k+\Delta\omega_0NT_0+q_{inj}\tilde\Gamma(\theta_k)$ (an extension of the discrete arithmetic in [P3] Sec. IV footnote 7)	Section 3	final (verified)
65	Noise corner of the first-order discrete-time loop ($=\Delta\omega_L$ at $\Delta\omega=0$)	$f_c=\dfrac{\beta}{1-\beta}\cdot\dfrac{f_{ref}}{2\pi}\approx\dfrac{\beta f_{ref}}{2\pi}$; exact discrete closed form $f_c'=\dfrac{f_{ref}}{2\pi}\arccos\!\big(1-\dfrac{\beta^2}{2(1+\beta)}\big)$	subharmonic_injection	derived on this site from the $\vert H\vert^2=1/2$ point of $H_{osc}(z)=(1-z^{-1})/(1-(1-\beta)z^{-1})$; agrees with the continuous-time $\omega_c=\sqrt{\omega_L^2-\Delta\omega^2}$ of injection_locking_noise for $f\ll f_{ref}$	Section 4.2	final (verified)
66	Closed-form output jitter (time-averaged phase variance)	$\sigma_{out}^2=\sigma_w^2\Big[\dfrac{(1-\beta)^2}{\beta(2-\beta)}+\dfrac12\Big]=\kappa^2NT_0\cdot\dfrac{1-\beta+\beta^2/2}{\beta(2-\beta)}$ ($\sigma_w^2=\kappa^2T_{inj}$; MC ratio 0.999)	subharmonic_injection	derived on this site from the linearized map's geometric series $\theta_k^-=\sum_j(1-\beta)^jw_{k-j}$; $\kappa^2$ is the variance-growth rate of [P2] Eq.(11), p.793 (verified)	Section 4.3	final (verified)
67	Generalized Adler equation (Mirzaei) and the large-injection lock range ([P4] Eq.(8)–(9))	$\dfrac{d\theta}{dt}=\omega_0-\omega_{inj}-\dfrac{\omega_0}{2Q}\dfrac{I_{inj}\sin\theta}{I_{osc}+I_{inj}\cos\theta}$ (Eq.8); $\omega_L=\dfrac{\omega_0}{2Q}\dfrac{I_{inj}}{I_{osc}}\dfrac{1}{\sqrt{1-I_{inj}^2/I_{osc}^2}}$ (Eq.9)	paper_004_large_injection_transient	paper_004 (Hong Part II 2019), Sec. III-A, p.2123 (verbatim transcription, verified)	Section 1.1	final (verified)
68	Effective ISF inversely proportional to amplitude ([P4] Eq.(13))	$\tilde\Gamma_{LC}=\dfrac{\tilde\Gamma}{1+A}$	paper_004_large_injection_transient	paper_004, Sec. III-C, p.2124 (verbatim transcription, verified)	Section 1.2	final (verified)

Legend

• final: the final result formula for that topic.
• step: an intermediate step in the derivation.
• reference: an external model used for comparison (not necessarily from the 5 downloaded PDFs).
• ⚠️: manual_verification_needed = true — please re-check against the original PDF.