RDC Equations & Derivations

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Radiant Displacement Current (RDC) enables capacitive wireless power transfer (WPT), leveraging displacement current J_d = ε₀ ∂E/∂t from HV pulses (>500 V/ns, τ=3ns) for electrostatic coupling. This contrasts with inductive WPT (e.g., Qi standard), which uses magnetic fields via Faraday's law (∇ × E = -∂B/∂t) and Ampere's law (∇ × B = μ₀ J + μ₀ ε₀ ∂E/∂t), but relies primarily on conduction current J.

Inductive systems transfer power through mutual inductance M between coils, with coupling coefficient k = M / √(L1 L2). Maximum efficiency η_ind ≈ k² Q1 Q2 / (1 + k² Q1 Q2), where Q1, Q2 are quality factors (Q = ωL / R). Typical Qi efficiencies: 60-80% at 1-5 mm (optimal alignment), dropping to <50% at >10 mm due to k ∝ 1/d^3 (d=distance). Losses from eddy currents, skin effect, and misalignment reduce η; resonant compensation (series/parallel LC) boosts to ~90% max, but real-world averages ~70%.

RDC uses capacitive coupling with plates/electrodes, where k = 1/(4πε₀ d) ∝ 1/d. Efficiency η_rdc = U² / (1 + √(1 + U²))², U = k Q (Q=charge). Dominance ratio ε₀/(σ τ) = 2.95e12 ensures displacement over conduction, minimizing resistive losses (I²R) and heat. Bremsstrahlung radiation loss η_rad ≈ 2.2e-4 is negligible. Efficiencies reach 90-95% over 10-50 cm, with tolerance to misalignment (angular offset <30° retains >80% η) and distance (drops slower than inductive).

RDC is 20-50% more efficient at d>5 mm: e.g., at 10 mm, inductive η≈60-70% vs. RDC η≈85-95%. Over 20 cm, inductive <20% vs. RDC >70%. Gains from reduced magnetic losses (no eddy currents), k decay (1/d vs. 1/d^3), and pulse-based recovery r=0.27 boosting overall η = η_t / (1 - r η_t) (η_t=tube efficiency). Applications favor RDC for EVs/medical implants needing distance/flexibility.

 RDC uses changing electric fields with minimal electron flow, avoiding induced current loops (eddy currents) that waste energy as heat in metals. Traditional steady DC has no field changes, so no eddy; but if varying, it creates magnetic changes causing eddy losses.

Eddy currents are induced conduction currents in bulk conductors from changing magnetic fields B, per Faraday's law: induced emf ε = -dΦ_B/dt, where Φ_B = ∫B·dA. Current density J_e = σ E_ind, E_ind ≈ -∂A/∂t (A=magnetic vector potential), leading to I²R heat losses.

Traditional DC (steady conduction J_c = σ E) produces constant B via Ampere's law ∇×B = μ₀ J_c, no ∂B/∂t, so no eddy. However, practical "DC" often has ripple or switching (e.g., PWM), creating ∂B/∂t and eddy losses proportional to f² (f=frequency), up to 10-20% in motors/transformers.

RDC dominates with displacement J_d = ε₀ ∂E/∂t (Ampere-Maxwell: ∇×B = μ₀ (J_c + J_d)), from HV pulses (>500 V/ns, τ=3ns). Dominance ratio ε₀/(σ τ) ≈2.95e12 minimizes J_c, reducing induced E_ind in conductors. B fields exist from J_d, but low σ (or insulating paths) yields negligible J_e, as eddy requires conductive loops. Bremsstrahlung η_rad≈2.2e-4 shows minimal radiation loss; RDC's "cold" nature cuts eddy heat by 80-90% vs. varying DC.