37-EFT.WP.EDX.HighSpeed v1.0 | Chapter 6 — Modes & Interconnect Primitives (S40-HF / I30-HF)

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

Chapter 6 — Modes & Interconnect Primitives (S40-HF / I30-HF)

I. Chapter Objectives & Structure

1. Objective: Provide the minimal expressions for modal decomposition and primitive composition of high-speed interconnects (S40-HF*), and define the layout–path–mode binding interfaces (I30-HF) to support in-place computation, recording, and verification of Z_eft(omega).
2. Structure: Symbols & domain → S40-HF modes & compositions → I30-HF binding & interfaces → 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: omega.
• Modes: even/odd, common/differential; mode set 𝓜 = {m}.
• Propagation: alpha_m(omega), beta_m(omega), k_m = alpha_m + i·beta_m, Z_c,m(omega).
• Ports: Sij(omega), Z_eft(omega); paths & weights: gamma(ell), d ell, w_p(omega), w_m(omega).
• Primitives: microstrip/stripline segments, bends/splits, vias/plane transitions, connectors/launches.
• Hard QA gates: check_dim = pass, passivity (Re{Z_eft} ≥ 0), KK_consistency = pass.

S40-HF — Modes & Interconnect Primitives (Minimal)

S40-HF-1 (Modal decomposition)
Given port vector x = [V; I], project onto modal basis M(omega)
x_m(omega) = M(omega) · x(omega)
with modal propagation
x_m(L,omega) = D_m(omega) · x_m(0,omega) ,
D_m(omega) = exp( − alpha_m · L ) · exp( − i · beta_m · L ).

S40-HF-2 (Path–mode weights)
For path family {γ_p} and mode set 𝓜, coherent in-window superposition
R_coh(omega) = | Σ_p Σ_{m∈𝓜} w_{p,m}(omega) · r_{p,m}(omega) · exp( i · omega · T_arr,p ) | ,
energy composition (out of window)
R_inc(omega) = ( Σ_p Σ_m w_{p,m}(omega) · | r_{p,m}(omega) |^2 )^{1/2} ,
with constraints w_{p,m} ≥ 0, Σ_p Σ_m w_{p,m} ≤ 1.

S40-HF-3 (Primitive-level composition)
For cascaded/branched segments {seg}, compose via transmission/ABCD matrices while preserving modal consistency:
T_total(omega) = Π_{seg} T_seg(omega) ,
where each T_seg is parameterized by alpha_m, beta_m, Z_c,m and records segment-level path gamma_seg and n_eff(seg).

S40-HF-4 (Mode conversion & scattering)
Discontinuities (bends/splits/steps) induce mode conversion via a coupling matrix C(omega)
x_m^+(out) = C(omega) · x_m^+(in) ,
with |C_{mn}| ∈ [0,1]. Update w_{p,m}(omega) accordingly and reflect any ΔT_arr.

S40-HF-5 (Leakage channels in vias & plane transitions)
If long stubs/residual stubs or unbridged plane transitions exist, introduce leakage paths γ_side, raising w_side(omega); keep consistency with radiation correction ΔZ_rad(omega) activation (see Chapter 5).

I30-HF — Layout–Path–Mode Binding (Binding & Interfaces)

I30-HF-1 (HF binding)
binding_hf = bind_layout_hf(layout, stackup, returns, constraints)
Outputs a record containing paths, modes, segments, and initial weights: {Paths, Modes, Segments, Weights0}.

I30-HF-2 (Mode projection & merge)
x_m = mode_project(dataset, M(omega)) ;
dataset' = mode_merge(x_m, rules) ,
to produce equivalent common/differential or primary/secondary combinations while preserving passivity/K–K.

I30-HF-3 (Path correction & arrival recording)
aligned = path_correct_hf(dataset', binding_hf, arrival) ,
with mandatory arrival{form,gamma,measure,c_ref,Tarr,u_Tarr,delta_form} and segment-level n_eff(seg).

I30-HF-4 (S↔Z & port-normalization consistency)
Z_eft = map_S_to_Z(S, Znorm(omega)) , explicitly stating Z_c(omega) and passing check_dim/passivity/KK.

III. Implementation & Records (minimum execution dialect)

• Required fields:
  binding_ref, Paths{γ_p, segments{len_m, n_eff, layer, neigh}}, Modes{m, Z_c,m, alpha_m, beta_m}, Weights{w_{p,m}}, arrival{...}, deemb, sync(Δt_sync), qa_gates{check_dim, passivity, KK}.
• Phase correction: arg Z_corr(omega) = arg Z_raw(omega) − ( omega · Δt_sync ).
• Arrival consistency: two-dialect T_arr difference ≤ u(T_arr).
• Path/mode robustness: ΔW = Σ_{p,m} | w_{p,m}(ω2) − w_{p,m}(ω1) | as in-band robustness KPI.

IV. Falsifiability Criteria (for S40-HF / I30-HF)

• J-HF-40-1 (Modal consistency): If mode_project → mode_merge produces Z_eft that fails passivity/K–K, reject the projection/merge rules.
• J-HF-40-2 (Path–arrival): Under switchable routing/return experiments, slope k_φ of Δarg Z(omega) must satisfy k_φ ≈ ΔT_arr; otherwise reject path modeling or binding records.
• J-HF-40-3 (Primitive composition): If Π T_seg deviates beyond gate vs measurement/3D-EM, reject segment alpha/beta/Z_c or boundary conditions.
• J-HF-40-4 (Leakage channel): After enabling γ_side, if trends of ΔW and Re{ΔZ_rad} disagree, reject channel annotation or coupling matrix.

V. Compliance Templates (copy-ready)

• Binding record (YAML)
• binding_hf:
• id: "LAY2PATH-HF-0001"
• ports: ["P1","P2"]
• paths:
• - id: "γ_main"
• segments:
• - {layer:"L3", len_m:0.0100, n_eff:2.12, neigh:"GND_L2"}
• - {layer:"L4", len_m:0.0020, n_eff:2.18, neigh:"GND_L5"}
• weight_init: 0.88
• - id: "γ_side"
• segments:
• - {layer:"L3", len_m:0.0035, n_eff:2.40, neigh:"slot_L2"}
• weight_init: 0.12
• modes:
• - {name:"DM", Zc_ohm:[...], alpha_per_m:[...], beta_per_m:[...]}
• - {name:"CM", Zc_ohm:[...], alpha_per_m:[...], beta_per_m:[...]}
• coupling:
• C:
• DM_to_CM: [ ... ] # |C_{mn}| ∈ [0,1]
• CM_to_DM: [ ... ]
• Arrival record
• arrival:
• form: "n_over_c" # or "one_over_c_times_n"
• gamma: "explicit"
• measure: "d_ell"
• c_ref: 299792458.0
• Tarr_s: 1.234e-09
• u_Tarr_s: 6.0e-12
• delta_form: "n_over_c"
• Numerical flow (pseudocode)
• # Mode projection
• x_m = M(ω) @ x_port(ω)
• # Segment propagation
• for seg in segments:
• for m in modes:
• x_m = x_m * exp(-alpha_m(seg,ω)*Lseg) * exp(-1j*beta_m(seg,ω)*Lseg)
• # Mode conversion
• x_m = C(ω) @ x_m
• # Path/mode superposition
• R_coh = abs(sum_p_m( w[p,m](ω) * r[p,m](ω) * exp(1j*ω*Tarr[p] ) ))
• # QA
• assert min(Re(Z_eft)) >= 0.0 and KK_consistency(Z_eft)

VI. Cross-Chapter Links & Closure

• Dependencies: Chapter 2 (Terms & Symbols), Chapter 3 (P10-HF Axioms), Chapter 4 (Minimal Equations & Dispersion), Chapter 5 (Radiation Correction).
• Downstream: Chapter 7 (Coherence-window KPIs & gates), Chapter 12 (Layout & Process Rules), Chapter 14 (SimStack & cases), and the implementation/inference flows of I30-HF / I40-HF / Mx-*.