Regime-Level Energy Accounting
in Nonlinear Electrodynamic Systems:
An Event–Frequency Interpretation Framework

§ 1 — Introduction

This paper defines a two-layer energy accounting framework for Armstrong-type nonlinear electrodynamic systems operating in a controlled discharge-resonant regime that redistribute energy through discrete internal regime events at high repetition frequency. At the complete device boundary, the framework imposes classical conservation: \(P_{\mathrm{in,boundary}} = P_{\mathrm{customer}} + P_{\mathrm{losses}} + \dfrac{dE_{\mathrm{stored}}}{dt}\). At the internal regime level, it organizes per-event redistribution across load, feedback, and loss channels, bridged to macroscopic average power by \(P_{x,\mathrm{avg}} = E_{x,\mathrm{event}} \cdot f\). Event-level quantities and boundary-level power balance describe distinct analytical layers of the same system and must not be conflated.

Nonlinear electrodynamic systems that operate in pulsed or regime-based modes — such as repetitive gas discharges, pulsed-power plasmas, and high-frequency streamer regimes — frequently exhibit dynamics that are not well captured by simple linear steady-state assumptions. In many experimental and evaluative contexts, attention is focused on the apparent energy associated with a single discharge or switching event, while the repetition frequency and duty cycle of these events are neglected or treated inconsistently. This practice often leads to systematic underestimation of achievable macroscopic power levels and to misinterpretation of device behavior, especially when events occur at kilohertz to megahertz rates.

Contemporary pulsed discharges and plasma-processing systems routinely operate with pulse repetition frequencies from kilohertz to tens of megahertz, and average powers ranging from watts to kilowatts. Experimental and modeling studies in high-repetition-rate discharges and pulsed laser–plasma interactions consistently demonstrate that the average power is determined by the product of the per-pulse (or per-event) energy and the repetition rate, with additional structure introduced by duty cycle, waveform shape, and loss channels.

The purpose of this work is to formalize an interpretation framework for analyzing observed operating behavior in regime-based electrodynamic systems, connecting event-level energy transfer, repetition frequency, and system-level power balance in a manner that is explicitly consistent with classical electrodynamics yet independent of any particular implementation. The framework emphasizes a two-layer description: a boundary level, where conventional conservation laws apply to the complete device, and a regime level, where internal discrete events redistribute energy among functional roles. The analysis clarifies the distinction between energy sourcing — which must be evaluated at the complete device boundary — and internal energy redistribution, which structures regime dynamics but does not itself define the net input power.

§ 3 — Event–Frequency Relation for Average Power

§ 3.1 — Discrete Event Representation

Consider a periodic or quasi-periodic sequence of discrete internal events with repetition frequency \(f\), such that events occur at times \(t_k = k/f\) for integer \(k\), and the energy associated with path \(x\) in the \(k\)-th event is \(E_{x,k}\). Over an observation interval \(T\) containing \(N = fT\) events, the total energy delivered through path \(x\) is:

The corresponding time-averaged power is:

If event-to-event variations are small, one can define a characteristic event energy \(E_{x,\mathrm{event}}\) as:

leading directly to the key bridge relation:

Equation (8) is the key bridge between microscopic (event-level) and macroscopic (average) descriptions, and it is the standard way to connect pulse energy, repetition rate, and average power in pulsed systems such as lasers and repetitive discharges [7, 8, 9].

For architectures in which events run over \(N_{\mathrm{ch}}\) parallel channels — in the VENDOR.Max architecture, \(N_{\mathrm{ch}} \geq 3\) parallel discharge channels with overlapping frequency spectra per patent claim 5 — the bridge relation generalizes to \(P_{x,\mathrm{avg}} = E_{x,\mathrm{event}} \cdot f \cdot N_{\mathrm{ch}}\), with the derivation above applying per channel. The channel count \(N_{\mathrm{ch}}\) is written with a subscript to distinguish it from the event count \(N = fT\) of equation (5).

§ 3.2 — Relation to Instantaneous Power Waveforms

An alternative representation begins from the instantaneous power waveform \(p_x(t) = v_x(t)\,i_x(t)\) associated with a given path. The energy per event is:

where \(\Delta t\) is the event duration, often much smaller than the period \(1/f\). For a perfectly periodic waveform, the time-averaged power over one period is:

The distinction between peak power during an event and time-averaged power is particularly important in systems where peak powers can reach very high levels while average power remains in the kilowatt range.

§ 5 — Electromagnetic Energy Transfer to an Extraction Circuit

§ 5.1 — Inductive Coupling and Faraday’s Law

In many practical realizations, energy stored in an internal electrodynamic regime is coupled to an extraction circuit via electromagnetic induction, capacitive coupling, or a combination thereof. For inductive coupling, the instantaneous electromotive force (EMF) induced in a coil of \(N\) turns linked by magnetic flux \(\Phi(t)\) is Faraday’s law in lumped form [1, 2]:

When the induced EMF is applied across a load with current \(i(t)\), the instantaneous power delivered to the load is:

The time-averaged power delivered to the load over an interval \(T\) is:

§ 5.2 — Consistency with Boundary-Level Energy Accounting

The inductive energy transfer described by (14)–(16) is a local manifestation of the global energy balance expressed by (1)–(3): changes in magnetic flux correspond to reconfiguration of electromagnetic field energy, and the product of EMF and current represents the rate at which field energy is converted to work on charges in the extraction circuit.

In the global picture, the Poynting vector flux across the device boundary equals the net power entering or leaving the system, while internal field reconfigurations — including inductive coupling to coils — redistribute energy among internal and external degrees of freedom. The complete device boundary power balance (2)–(3) remains the authoritative statement regarding overall energy conservation; the sum over all events and all paths is constrained by the total net input.

§ 5.3 — Measurement and Boundary-Level Verification

Correct evaluation of nonlinear regime-based electrodynamic systems requires measurement at the complete device boundary, not at internal nodes or sub-circuits. Internal nodes carry quantities that belong to Layer 2 (regime-level redistribution) and cannot, on their own, characterize the Layer 1 input–output balance. The authoritative reference for testing energy conservation is the enclosing surface defined in equations (1)–(3): \(P_{\mathrm{in,boundary}}\) referenced at the complete device boundary as an aggregate accounting quantity, and \(P_{\mathrm{customer}}\) at the external load interface.

At the complete device boundary, the boundary-crossing inventory comprises all physical channels: conducted electrical terms, thermal flows, radiative losses, and field-mediated interaction with the surrounding medium. Boundary closure is verified against the complete inventory — not against the electrical port list alone.

At regime-level operation with high internal repetition frequency, instantaneous voltage and current waveforms at any internal node can exhibit non-sinusoidal, sharply discretized features. These waveform features describe regime structure; they do not, by themselves, define power. Time-averaged power is recovered through the integral \(P = \langle v(t)\,i(t)\rangle\) over an interval much longer than the event period \(1/f\), or equivalently through the bridge relation \(P = E_{\mathrm{event}} \cdot f\) defined in equation (8). Reading peak values directly as power, or comparing instantaneous waveform amplitudes between internal nodes and the complete device boundary, is a category error: peak power and time-averaged power are distinct quantities, and only the latter participates in the boundary balance.

Consequently, no measurement at an internal node — the regulated feedback path, the capacitive node, or any intermediate winding terminal — can replace boundary-level measurement of \(P_{\mathrm{in,boundary}}\) and \(P_{\mathrm{customer}}\). Internal measurements characterize regime organization; boundary measurements characterize energy accounting. Both are meaningful within their respective layers, but only the latter answers the conservation question. Verification of overall energy balance is performed exclusively at the complete device boundary under time-averaged conditions consistent with equations (2)–(3) and (8).

§ 7 — Interpretation Principles

The two-layer framework yields four principles essential for correct interpretation of experimental data and system behavior in nonlinear, regime-based electrodynamic systems.

Failure to distinguish these layers leads to erroneous comparisons between event energy and continuous power, to apparent contradictions with conservation of energy, and to incorrect extrapolations of experimental results.

§ 7.5 — Interpretation Boundaries and Common Failure Modes

Beyond the four principles of § 7, five specific analytical failure modes recur in evaluations of regime-based electrodynamic systems. Each failure mode shares a common structural origin: implicit conflation of analytical layers. The two-layer framework defined in § 2 makes this conflation explicit and corrigible. These interpretation boundaries are not additional physical assumptions but constraints on correct analytical mapping between already defined layers.

All five failure modes share a common structural origin: implicit layer conflation. Each is corrected by the same procedural discipline — identify which analytical layer the question belongs to, then apply the correct formula for that layer. The framework provides this procedural discipline; the failure modes diagnose what happens when it is omitted.

§ 7.6 — Energy Transformation Map: Stage × Quantity

The interpretation principles of § 7 and the failure modes of § 7.5 can be compressed into a single map. The rows below follow the eight-stage patent sequence of the VENDOR.Max architecture, documented in full at How VENDOR.Max Works; this article does not restate the stage mechanics — it maps which physical quantity is calculable at which stage. Quantities live at different analytical levels — complete device boundary (macroscopic accounting), per-event partition (regime domain), and gap-internal carrier dynamics — and carry different units. The carrier-dynamics column refers to the Townsend pre-breakdown forms of the patent-baseline analytical framework (ES2950176B2 family, priority 2023); the engineering implementation has advanced beyond that baseline, and the microscopic mechanism of the sealed switching cells is not disclosed. Boundary closure in the final column is verified through the conservation closure residual R_boundary — the requirement that the terms of equation (2) sum to zero residual within measurement uncertainty under accredited metrology.

§ 9 — Conclusion

A two-layer interpretation model for nonlinear, regime-based electrodynamic systems has been developed, connecting discrete energy-redistribution events with macroscopic power output through a frequency-domain perspective anchored in classical electrodynamics. At the complete device boundary, standard conservation laws enforce energy accounting and define the net power balance, while internally, event-level relations describe how event-associated energy is partitioned among load, feedback, and irreversible loss channels.

By formalizing the relation \(P_x = E_{x,\mathrm{event}} \cdot f\) — with its parallel-channel generalization \(P_{x,\mathrm{avg}} = E_{x,\mathrm{event}} \cdot f \cdot N_{\mathrm{ch}}\) — and embedding it in a consistent boundary-level energy balance, the framework eliminates a common source of misinterpretation in the evaluation of pulsed and regime-based systems — namely, the direct comparison of event energy to continuous power without accounting for event frequency. The illustrative example shows explicitly how millijoule-scale events at multi-megahertz internal regime frequencies correspond to kilowatt-scale average power delivery, fully within the bounds of classical energy conservation.

This interpretation framework is intended as a tool for analyzing and communicating experimental results in nonlinear electrodynamic systems, providing a mathematically consistent and physically transparent connection between internal regime dynamics and system-level performance. It provides the scientific foundation for the VENDOR.Max platform — an Armstrong-type nonlinear electrodynamic oscillator at validation stage TRL 5–6 — while remaining independent of implementation-specific disclosure, protected design details, and proprietary operating parameters.

Beyond defining the relation between event-level energy and macroscopic power, the framework establishes strict interpretation boundaries preventing systematic category errors, including: (i) event-to-power miscomparison without frequency aggregation, (ii) feedback misclassification as net energy source, (iii) startup-impulse misinterpretation as steady-state input, (iv) interaction-medium misclassification as energy source, and (v) misreading of the boundary equation as a supply-topology statement rather than an accounting constraint at the complete device boundary. Each boundary is enforced by the two-layer accounting structure and its canonical bridge relation \(P_x = E_{x,\mathrm{event}} \cdot f\).

The present framework defines the correct interpretation model for regime-based electrodynamic systems. It does not, by itself, constitute experimental proof of any specific implementation. Empirical validation of operating performance is addressed separately through controlled testing, time-averaged measurements at the complete device boundary in the sense of § 5.3, and independent third-party verification protocols associated with the certification pathway toward TRL 7–8. The framework and the empirical validation programme are complementary but analytically distinct: this article specifies the language of correct evaluation; the validation programme specifies the data.