RDC Equations & Derivations

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Radiant Displacement Current (RDC) supports capacitive wireless power transfer (WPT) by using displacement current J_d = ε₀ ∂E/∂t from fast high-voltage pulses (>500 V/ns, rise time τ=3 ns). This creates strong electrostatic coupling between electrodes (like the tube's grids and a nearby receiver), with energy flowing via surrounding electric fields.

Inductive WPT (e.g., Qi standard) relies mainly on magnetic fields and conduction current, following Faraday's law (∇ × E = -∂B/∂t) and Ampère's law (∇ × B = μ₀ J + μ₀ ε₀ ∂E/∂t).

Inductive WPT EfficiencyPower transfers through mutual inductance M between coils. 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: 60–80% at 1–5 mm (good alignment); drops below 50% beyond 10 mm because k ∝ 1/d³. Losses come from eddy currents, skin effect, and misalignment. Resonant compensation can reach ~90% max, but real-world average is ~70%.

RDC Capacitive WPT EfficiencyUses capacitive coupling between plates/electrodes; coupling k ∝ 1/d (approximated as 1/(4πε₀ d) for simple cases). Efficiency: η_rdc = U² / (1 + √(1 + U²))², where U = k Q. The dominance ratio ε₀ / (σ τ) ≈ 2.95 × 10¹² (air: σ ≈ 3 × 10⁻¹⁵ S/m, τ=3 ns) ensures displacement current dominates, greatly reducing resistive I²R losses and heat. Bremsstrahlung radiation loss η_rad ≈ 2.2 × 10⁻⁴ is negligible. Pulse recovery factor r=0.27 further boosts overall efficiency: η = η_t / (1 - r η_t) (η_t = tube transfer efficiency). Potential advantages: better distance tolerance and misalignment (angular offset <30° can retain >80%); efficiency drops slower than inductive due to 1/d vs. 1/d³ decay.

Comparison

• At distances >5 mm: RDC can potentially achieve 20–50% higher efficiency (e.g., at 10 mm inductive ~60–70% vs. RDC higher).
• At 20 cm: inductive often <20%; RDC retains more. Key gains: no eddy currents, reduced magnetic losses, field-based transfer. Ideal for applications like EVs or medical implants needing greater distance or flexibility. Results depend on setup—use shielding/filters to manage EMI. These are based on prototype behavior and theory; real performance varies with testing.

 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.