37-EFT.WP.EDX.HighSpeed v1.0 | Chapter 4 — Minimal Equations & Dispersion Relations (S20-HF / S30-HF)

Home ／ Docs-Technical WhitePaper ／ 37-EFT.WP.EDX.HighSpeed v1.0

Chapter 4 — Minimal Equations & Dispersion Relations (S20-HF / S30-HF)

I. Chapter Objectives & Structure

1. Objective: State the high-frequency minimal equations S20-HF-* and dispersion relations S30-HF-*, linking phase, group delay, and propagation constants under a unified dialect, and define engineering records and release gates.
2. Structure: Symbols & domain → S20-HF minimal relations → S30-HF dispersion relations → Implementation & records → Falsifiability → Compliance templates → Cross-chapter closure.
3. Shared time-of-arrival dialect (equivalent; explicit gamma(ell) and d ell; record delta_form):
   • Constant-factored: T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
   • General: T_arr = ( ∫ ( n_eff / c_ref ) d ell )

II. Symbols & Domain

• Frequency domain: omega.
• Propagation & dispersion: alpha(omega) (Np·m⁻¹), beta(omega) (rad·m⁻¹), k(omega)=alpha+i·beta.
• Ports & characteristics: Z_eft(omega), Z_ref(omega), Z_c(omega), Sij(omega).
• Paths & weights: gamma(ell), d ell, w_p(omega), ΔT_arr.
• Phase & group delay: arg Z_eft(omega), T_group(omega)=d/domega( arg Z_eft ).
• Radiation & positive-real correction: ΔZ_rad(omega) (require Re{ΔZ_rad} ≥ 0).
• Hard QA gates: check_dim = pass, passivity (Re{Z_eft} ≥ 0), KK_consistency = pass.

S20-HF — Minimal Relations

S20-HF-1 (Group-delay definition)
T_group(omega) = d/domega ( arg Z_eft(omega) )
Domain: coherence window; phase synchronized as arg Z_corr = arg Z_raw − ( omega · Δt_sync ).

S20-HF-2 (Linear-phase approximation)
Within the coherence window,
arg Z_eft(omega) ≈ ( omega · T_arr ) + φ_T(omega),
with slowly varying φ_T(omega) and bounded second-order curvature (see Chapter 7 E_phase gate).

S20-HF-3 (Path-form dispersion leading term)
For dominant path length L,
beta(omega) · L ≈ omega · T_arr + φ_T(omega).
Hence the consistency relation:
T_group(omega) ≈ T_arr + dφ_T(omega)/domega ≈ L · ( dβ(omega)/domega ).

S20-HF-4 (S↔Z mapping & port normalization)
Mapping multiport Sij to Z_eft must preserve passivity and causality; port normalization impedance and Z_c(omega) deviations must be explicitly recorded and pass KK_consistency.

S20-HF-5 (Two-dialect arrival consistency)
For the same gamma(ell) and n_eff,
| T_arr^{(n_over_c)} − T_arr^{(n_over_c_times_n)} | ≤ u(T_arr),
otherwise no release.

S30-HF — Dispersion Relations

S30-HF-1 (Propagation constant & group delay)
k(omega) = alpha(omega) + i·beta(omega)
T_group(omega) ≈ L · dβ/domega consistent with S20-HF-3; engineering gate requires
| T_group − ( L · dβ/domega ) | ≤ u_Tgroup.

S30-HF-2 (Amplitude dispersion & loss)
Under matching and single-mode approximation,
|S21(omega)| ≈ exp( − alpha(omega) · L ).
alpha(omega) decomposes into dielectric, conductor (skin), and radiation loss; if the radiation channel is enabled, include it in ΔZ_rad(omega) and ensure Re{ΔZ_rad} ≥ 0.

S30-HF-3 (Minimal mapping for material/conductor dispersion)

• Dielectric effective index: n_eff(omega) ⇒ beta(omega) = ( n_eff(omega) / c_ref ) · omega (first order).
• Conductor effective conductivity: sigma_eff(omega) affects alpha(omega) and Z_c(omega); apply band-limited/smooth priors and forbid non-physical oscillatory poles (right half-plane).

S30-HF-4 (Out-of-window composition rule)
Outside the window, do not use the linear-phase approximation; compute response via energy composition:
R_inc(omega) = ( Σ_p w_p(omega) · |r_p(omega)|^2 )^{1/2},
and explicitly label windows and gates.

IV. Implementation & Records (minimum execution dialect)

1. Required records:
   arrival{form,gamma,measure,c_ref,Tarr,u_Tarr,delta_form}, binding_ref, deemb, sync(Δt_sync), Z_c(omega), alpha_per_m, beta_per_m, T_group_s, E_phase, GDR, and (if enabled) ΔZ_rad(omega).
2. Consistency checks:
   T_group vs L·dβ/dω; 2) two-dialect T_arr; 3) passivity / KK_consistency.
3. Record template:
4. highspeed:
5. band_GHz: [f_min, f_max]
6. coherence_window: {w1: ω1, w2: ω2}
7. arrival: {form:"n_over_c", gamma:"explicit", measure:"d_ell", c_ref:299792458.0,
8. Tarr_s:..., u_Tarr_s:..., delta_form:"n_over_c"}
9. dispersion:
10. alpha_per_m: [...]
11. beta_per_m: [...]
12. T_group_s: [...]
13. Zc_ohm: [...]
14. radiation_gate:
15. Re_Zrad_min: 0.0
16. KK_consistency: "pass"
17. qa_gates: {check_dim:"pass", passivity:"pass", KK:"pass"}

V. Falsifiability (for S20-HF / S30-HF)

• J-HF-S20-1: In-window linear-phase fit residual exceeds E_phase or shows systematic second-order curvature → reject the S20-HF-2 window.
• J-HF-S20-2: T_group vs L·dβ/dω difference exceeds u_Tgroup → reject S20-HF-3 / S30-HF-1 consistency or the β estimate.
• J-HF-S20-3: Two-dialect T_arr exceeds u(T_arr) → reject S20-HF-5 or path/units/records.
• J-HF-S30-1: After sealing gaps/shortening stubs, if Re{ΔZ_rad} and |E_rad|/|I_CM| do not decrease, reject the radiation equivalence mapping.
• J-HF-S30-2: If alpha(omega) fit yields Re{Z_eft}<0 or K–K fails, reject the loss model or the S↔Z mapping workflow.

VI. Compliance Templates (copy-ready)

• Numerical sketch (pseudocode):
• # Phase after sync correction
• phi = argZ_corr[ω1:ω2] # coherence window
• # Group delay
• T_group = grad(phi, omega)
• # Phase linearity error & group-delay ripple
• E_phase = max_abs(phi - (omega*Tarr + phi0_opt))
• GDR = max_abs(T_group - median(T_group))
• # Propagation-constant consistency
• beta = unwrap(phi) / L # approx; or extract from transmission/simulation
• T_group_beta = L * grad(beta, omega)
• assert max_abs(T_group - T_group_beta) <= u_Tgroup
• # Amplitude dispersion (matching approx)
• alpha = -log(abs(S21)) / L
• # Hard gates
• assert min(Re(Z_eft)) >= 0.0 and KK_consistency(Z_eft)
• Two-dialect arrival gate:
• Tarr_dual:
• diff_s: |Tarr_n_over_c - Tarr_n_over_c_times_n|
• u_Tarr_s: ...
• pass: (diff_s <= u_Tarr_s)

VII. Cross-Chapter Links & Closure

• Dependencies: Chapter 2 (Terms & Symbols), Chapter 3 (P10-HF Axioms), Current Chapter 8 (Paths/Arrival), Chapter 9 (Metrology Chain).
• Downstream: Chapter 5 (S50-HF radiation correction), Chapter 7 (coherence-window KPIs & gates), Chapter 12 (layout & process rules), Chapter 14 (SimStack & cases), Chapter 16 (design protocol & checklist).