38-EFT.WP.EDX.EMI v1.0 | Chapter 8 — S↔Z & Field Mapping (I40-EMI)

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Chapter 8 — S↔Z & Field Mapping (I40-EMI)

I. Chapter Objectives & Structure

1. Objective: Specify the canonical mapping from multiport S-parameters to tension-landscape impedance Z_eft(omega) for emission/immunity, together with mixed-mode (DM/CM) normalization; establish the field-mapping dialect from ΔZ_rad(omega) to near/far-field envelopes and I_CM(omega); fix record fields and release QA.
2. Structure: Symbols & domain → I40-EMI mapping & normalization → Port→radiation equivalence & field conversion → Implementation & records → Falsifiability → Compliance templates → Cross-chapter closure.
3. Shared time-of-arrival dialect (two equivalent forms; explicit gamma(ell) & 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 & ports: omega, S(omega) ∈ ℂ^{N×N}, Znorm(omega), Z_eft(omega), Z_c(omega).
• Mixed-mode: T_mm (SE→MM transform), Z0_mm (DM/CM normalization impedances), S_mm, Z_mm.
• Radiation & fields: ΔZ_rad(omega) (require Re{ΔZ_rad} ≥ 0), P_rad(omega), E_rad(omega,r), H_rad(omega,r), I_CM(omega).
• Paths & weights: gamma(ell), d ell, w_{p,m}(omega).
• Hard QA: check_dim = pass, passivity (Re{Z_eft} ≥ 0), KK_consistency = pass, two-dialect T_arr agreement.

I40-EMI — S↔Z Mapping & Mixed-Mode Normalization (Canonical)

I40-EMI-1 (N-port S→Z mapping)
Let Z0 = Znorm(omega) and I identity,
Z_eft(omega) = Z0^{1/2} · ( I + S(omega) ) · ( I − S(omega) )^{-1} · Z0^{1/2}
Requirements: (I−S) invertible; post-mapping must pass passivity/KK; record Znorm(omega) (array or constant).

I40-EMI-2 (Mixed-mode normalization & back-projection)
From single-ended S_se via T_mm: S_mm = T_mm · S_se · T_mm^{-1}, then normalize with Z0_mm:
Z_mm = Z0_mm^{1/2} · ( I + S_mm ) · ( I − S_mm )^{-1} · Z0_mm^{1/2}
Preserve power & reciprocity before back-projecting to SE; record versions/sources of T_mm/Z0_mm.

I40-EMI-3 (Port phase–arrival dialect)
In-window: arg Z_eft(omega) ≈ omega·T_arr + φ_T(omega); phase must be sync-corrected first:
arg Z_corr(omega) = arg Z_raw(omega) − ( omega · Δt_sync ) , then unwrap and fit.

Port→Radiation Equivalence & Field Mapping

I40-EMI-4 (Port→radiated power equivalence)
P_rad(omega) ≈ (1/2) · Re{ΔZ_rad(omega)} · |I_port(omega)|^2 ,
with ΔZ_rad(omega) = Z_eft(omega) − ( Z_ref(omega) + ΔZ_T(omega) ) and Re{ΔZ_rad} ≥ 0.

I40-EMI-5 (Near/Far-field envelope conversion)
At distance R with known antenna/probe factors:
|E_rad(omega,R)| ≈ K_E(omega,R) · √P_rad(omega) , |H_rad(omega,R)| ≈ K_H(omega,R) · √P_rad(omega)
K_E/K_H come from versioned calibration files; detector detector∈{pk,qpk,avg} IF bandwidth/time constants must be logged with the data.

I40-EMI-6 (Common-mode current consistency)
If CM port is represented by LISN/injection networks:
I_CM(omega) = H_cm(omega) · u_inj(omega) ,
its trend is monotonically consistent with Re{ΔZ_rad}(omega) and P_rad(omega).

III. Implementation & Records (minimum execution dialect)

1. Required fields:
   S(omega), Znorm(omega), (if used) T_mm/Z0_mm, Z_eft(omega)/argZ, Z_c(omega), ΔZ_rad(omega) with Re_Zrad_min, I_port(omega) (or equivalent), field-calibration & distance info for E_rad/H_rad, I_CM(omega) (if collected), binding_ref, arrival{form,gamma,measure,c_ref,Tarr,u_Tarr,delta_form}, qa_gates{check_dim,passivity,KK}.
2. Consistency checks:
   • Post-mapping Re{Z_eft} ≥ 0 and KK = pass;
   • P_rad vs |E_rad|/|H_rad| monotonic trends;
   • Two-dialect T_arr agreement;
   • Mixed-mode back-projection conserves power/reciprocity.

IV. Falsifiability (for I40-EMI)

• J-EMI-8-1 (Positive-real & causality): if Re{Z_eft} < 0 or KK fails, reject normalization/mapping or port basis.
• J-EMI-8-2 (Port–field inconsistency): if P_rad disagrees in trend with |E_rad|/|H_rad|, reject power equivalence or field probe/antenna setup.
• J-EMI-8-3 (Mixed-mode loop): if the loop S_se→S_mm→Z_mm→Z_se breaks power or reciprocity, reject T_mm/Z0_mm or the back-projection step.
• J-EMI-8-4 (Two-dialect agreement): if |T_arr^{(1)}−T_arr^{(2)}| > u(T_arr), reject release.

V. Compliance Templates (copy-ready)

• Mapping record (YAML)
• i40_emi_mapping:
• band_GHz: [f_min, f_max]
• sparams: "/artifacts/S.s2p"
• Znorm_ohm: [50.0, 50.0]
• mixed_mode:
• enabled: true
• T_mm: "/cfg/T_mm.yaml"
• Z0_mm_ohm: [100.0, 25.0]
• 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"
• impedance:
• Z_eft: {real:[...], imag:[...]}
• deltaZ_rad: {Re_ohm:[...], Im_ohm:[...]} # Re ≥ 0
• currents_fields:
• I_port_A: [ ... ]
• I_CM_A: [ ... ] # optional
• E_rad_peak_dBuV_m: [ ... ]
• H_rad_peak_dBA_m: [ ... ]
• qa_gates: {check_dim:"pass", passivity:"pass", KK:"pass"}
• Numerical flow (pseudocode)
• # S→Z mapping & mixed-mode normalization
• S_mm = T_mm @ S_se @ inv(T_mm)
• Z_mm = Z0_mm**0.5 @ (I+S_mm) @ inv(I-S_mm) @ Z0_mm**0.5
• Z_se = back_to_single_ended(Z_mm, T_mm) # if needed
• assert min(Re(Z_se)) >= 0.0 and KK_consistency(Z_se)
• # Port→field equivalence
• ΔZ_rad = Z_se - (Z_ref + ΔZ_T)
• P_rad = 0.5 * Re(ΔZ_rad) * abs(I_port)**2
• E_env = K_E(ω,R) * sqrt(P_rad); H_env = K_H(ω,R) * sqrt(P_rad)
• assert corr(P_rad, E_env) > rho_gate and corr(P_rad, H_env) > rho_gate
• # Arrival two-dialect agreement
• assert abs(Tarr_n_over_c - Tarr_one_over_c_times_n) <= u_Tarr

VI. Cross-Chapter Links & Closure

• Dependencies: Chapter 2 (Terms & Symbols), Chapter 3 (P10-EMI Axioms), Chapter 4 (Minimal Equations & Equivalents), Chapter 6 (Cables/Connectors & Mixed-Mode), EDX.HighSpeed Chapter 8 (S↔Z Mapping).
• Downstream: Chapter 9 (Metrology & Compliance Dialect — use this mapping to emit EMI fields), Chapter 10 (Experimental Design & Falsification — port–field consistency criteria), Chapter 11 (Engineering Rules: Shielding/Ground/Returns), and the appendices (implementation APIs/dataset cards) for release & regression.