RDC Equations & Derivations

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Traditional circuit design focuses entirely on Conduction Current Density (J_c), which represents the physical drift velocity of free electrons passing through a metallic lattice. This conventional framework inevitably experiences thermal dissipation due to atomic friction, governed by Joule's Law:

• P_loss = (I^2) * R

Our architecture alters this relationship by optimizing for the transient Displacement Current Density (J_d) term found in the Ampere-Maxwell law. By restricting the operational pulse-width to sub-microsecond timelines, the electric field changes at velocities faster than the relaxation time of the conductor.

The time-varying electric field propagates through the system as a localized spatial wave. Because the wave profile terminates before free electrons can drift and collide with the metal lattice, energy is transferred through the dielectric space without triggering typical thermal losses or short-circuit failures.

This section maps the mathematical modeling of high-dV/dt solid-state fields. All variables conform to standard IEEE and SI notation protocols where E_0 is the permittivity of free space (8.85 * 10^-12 F/m), Sigma is the medium conductivity, and Tau is the pulse rise time.

• 1. Displacement Current Density (J_d)
  Assessing the instantaneous field density under hyper-fast switching criteria:
• J_d = E_0 * (dE / dt)
• Baseline Constraint: Enabled via Silicon Carbide (SiC) switching channels executing edge-slopes where dV/dt exceeds 40 Volts per nanosecond.
• 2. The Dominance Ratio (J_d / J_c)
  To prove the structural dominance of the displacement field over standard conduction current within the dielectric switching gap:
• J_d / J_c = E_0 / (Sigma * Tau)
• Result: Utilizing an air/dielectric gap configuration where Sigma is roughly 3 * 10^-15 S/m and Tau is constrained to a 10 ns rise-time window yields a field dominance ratio exceeding 2.95 * 10^12. This confirms displacement current completely dominates the localized vector space.
• 3. Effective System Gain (COP_sys)
  The mathematical model tracking the recycling of reactive field echoes back to the source battery via parallel Schottky recovery networks:
• COP_sys = Core_Efficiency / (1 - (Recovery_Ratio * Core_Efficiency))
• Empirical Validation: Under standard benchmarks where Core_Efficiency = 0.85 and Recovery_Ratio = 0.27, the network yields a COP_sys of 1.10, representing a 10.3% operational runtime extension without violating thermodynamics.