Regime-Based Electrodynamic Systems as an Alternative to Linear Energy Models: Scientific Foundation of VENDOR.Energy™ Architecture
Authors: O. Krishevich, V. Peretyachenko
Abstract
This article provides a rigorous energy-balance framework for the VENDOR.Energy™ architecture — a class of nonlinear electrodynamic systems with functional separation between regime formation, loss compensation, and useful power extraction.
The central objective is to formalize the role of internal feedback power routing and buffering in maintaining the operating regime, under a boundary-consistent first-law accounting. Specifically, we clarify how the system routes a portion of the device-internal DC-bus power back to the active core (Circuit A). The DC bus is an internal distribution node fed by the external input and, where applicable, conditioned internal power from the extraction stage and/or the buffer; any such “conditioned internal power” denotes internal conversion and redistribution of energy already accounted for within the Device Boundary, not an additional independent input term. Therefore, the feedback path is strictly an internal power-allocation mechanism within the Device Boundary. We show why this closed-loop architecture is fully consistent with conservation laws when the system boundary is correctly defined.
The work shows that apparent “efficiency > 100%” conclusions arise exclusively from boundary-definition or measurement-incompleteness errors, not from any violation of physics.
Keywords: regime-based systems, nonlinear electrodynamics, pulsed resonance, gas discharge, energy balance, open systems, feedback architecture.
1. The Problem: Where Does the Sustaining Power Come From?
1.1 Why This Question Arises
In the VENDOR architecture, the active core (Circuit A) operates in a nonlinear electrodynamic regime with high internal energy circulation. This regime requires continuous compensation of irreversible losses — ohmic, dielectric, radiative, and discharge losses — to remain stable.
An observer examining Circuit A in isolation sees: a small maintenance power sustaining a regime that delivers a much larger power to the extraction circuit. The natural reaction is: where is the missing energy coming from?
This confusion has a precise origin: the observer is drawing the system boundary around the wrong subsystem.
1.2 The Answer in One Paragraph
The VENDOR system operates as a closed-loop architecture with two functionally separated circuits:
- Circuit A (Active Core) forms and maintains the nonlinear electrodynamic regime.
- Circuit B (Linear Extraction) extracts power from Circuit A via classical electromagnetic induction.
A fraction of the device-internal DC-bus power is allocated back through the regulated bus to Circuit A as maintenance power. The DC bus is an internal distribution node that may be fed by the external input and/or conditioned internal power paths (including the extraction stage and the buffer); here, “conditioned internal power” refers to internal redistribution of energy already within the Device Boundary, not a second external source. The feedback therefore remains an internal allocation within the Device Boundary. In steady operation, the DC bus is ultimately sustained by \(P_{\text{in,ext}}\), with transient deviations governed solely by \(\Delta E_{\text{stored}}\). The buffer battery with BMS regulates this allocation, smoothing transients and protecting regime stability.
Crucially, the feedback path does not replace external input. At the Device Boundary, any sustained operation with nonzero \(P_{\text{load}}\) requires nonzero time-averaged external input \(P_{\text{in,ext}}\), except during intervals where stored energy is being depleted. The feedback loop is a power-routing mechanism that allocates part of the DC-bus power back to the core; time-averaged net energy is accounted for by \(P_{\text{in,ext}}\) (with short-term deviations governed by \(\Delta E_{\text{stored}}\)), while the external supply accounts for the net balance of load + irreversible losses + storage change. In time average, any net energy delivered to the load and dissipated as losses must be supplied by \(P_{\text{in,ext}}\); internal recirculation can only redistribute energy and temporarily borrow/return it via \(\Delta E_{\text{stored}}\).
From the perspective of the complete device boundary, the feedback power is an internal redistribution — not a new energy source. The only true input is the external electrical power crossing the device boundary. The only outputs are useful load power and irreversible losses.
2. System Architecture and Energy Flow
2.1 The Two Circuits
Circuit A — Regime Formation (Active Core)
Circuit A is a nonlinear resonant structure based on an effective LC combination with gas discharge as a controlled nonlinear element. The effective resonant frequency is:
\[\omega_0 = \frac{1}{\sqrt{LC}}\]
In nonlinear regimes, \(\omega_0\) may depend on amplitude, conductivity, and discharge parameters; the value above is understood as the equivalent resonant frequency for the chosen operating point.
For a chosen equivalent loss model, an effective quality factor \(Q_{\text{eff}}\) is defined experimentally. For a series RLC representation, \(Q_{\text{series}} = \omega_0 L / R_s\); for a parallel representation, \(Q_{\text{parallel}} = \omega_0 R_p C\). In this work we use \(Q_{\text{eff}}\) as the measured regime quality factor at the chosen boundary.
The discharge provides dynamic nonlinear conductivity \(\sigma(E,t)\), enabling the system to reach and sustain a stable limit-cycle regime. This regime maintains high internal energy circulation with comparatively small maintenance power — a direct consequence of a high effective quality factor \(Q_{\text{eff}}\).
Key physics: High \(Q_{\text{eff}}\) means energy shuttles between electric and magnetic storage many times before being dissipated. The maintenance power only needs to compensate the fraction lost per cycle, not re-create the entire circulating energy.
In this paper, “circulation” denotes internal energy exchange and storage within the regime (fields/currents), not an additional external power inflow.
Circuit B — Linear Power Extraction
Circuit B operates on classical Faraday induction:
\[\mathcal{E} = -\frac{d\Phi_B}{dt}\]
The time-varying magnetic flux generated by Circuit A’s regime induces EMF in an extraction winding. This EMF is rectified, filtered, and converted to useful DC or AC output.
Lenz’s law applies fully: extraction reduces the loaded quality factor:
\[\frac{1}{Q_{\text{eff,loaded}}} = \frac{1}{Q_{\text{core}}} + \frac{1}{Q_L}\]
This additive relation is used here as an equivalent loss-partition model at a fixed definition of \(E_{\text{stored}}\); in practice, \(Q_{\text{eff,loaded}}\) is identified from measurement under the given operating regime.
Increased extraction → increased effective losses → increased maintenance power requirement.
2.2 The Feedback Loop
The critical architectural feature is the feedback path:
┌─────────────────────────────────────────────┐
│ DEVICE BOUNDARY │
│ │
P_in,ext ───────►│ ┌──────────┐ induction ┌──────────┐ │
(startup + │ │Circuit A │ ──────────────► │Circuit B │ │
external input) │ │(Active │ Faraday law │(Linear │ │
│ │ Core) │ ◄────────────── │Extraction│ │──► P_load
│ └────┬─────┘ feedback via └─────┬────┘ │
│ │ regulated DC bus │ │
│ │ ▲ │ │
│ │ │ │ │
│ │ ┌────┴─────┐ │ │
│ └────────►│ Buffer │◄───────┘ │
│ │ + BMS │ │
│ └──────────┘ │
│ │
└─────────────────────────────────────────────┘
▼
B_total (heat,
radiation, etc.)
What happens step by step:
- Startup: External power \(P_{\text{in,ext}}\) ignites the regime in Circuit A and charges the buffer.
- Regime formation: Circuit A reaches a stable nonlinear regime (limit cycle) with high internal energy circulation.
- Extraction: Circuit B extracts power from Circuit A’s regime via induction.
- Feedback: A portion of the device-internal power available on the regulated DC bus is allocated back as maintenance power to Circuit A. The DC bus may be supported by the external input and/or conditioned internal paths (including the extraction stage and the buffer), so this feedback remains an internal allocation within the Device Boundary; the device-level balance remains set by external input. In steady operation, the DC bus is ultimately sustained by \(P_{\text{in,ext}}\), with transient deviations governed solely by \(\Delta E_{\text{stored}}\).
- Regulation: The buffer + BMS smooths this feedback, compensating transients and load variations.
- Steady operation: The system operates as an internally regulated loop: the internal feedback path routes a portion of the DC bus power to maintain the regime, while the external input at the Device Boundary supplies the net energy required by (losses + delivered load power + storage change). The feedback is a power-routing mechanism, not an independent energy source.
2.3 The Buffer and BMS Role
The buffer battery is not a hidden energy source. It performs:
- Transient smoothing: absorbs load spikes and compensates regime disturbances
- DC bus stabilization: prevents voltage sags that could collapse the nonlinear regime
- Startup energy storage: provides initial ignition power before the feedback loop is established
- BMS intelligence: manages charge/discharge cycles, protects against over-extraction, and controls soft start/stop sequences
In steady state, the buffer’s net energy change averages to zero. Any energy withdrawn during transients is replenished from the DC-bus; the time-averaged net energy is accounted for by \(P_{\text{in,ext}}\) (with short-term deviations governed by \(\Delta E_{\text{stored}}\)).
3. Energy Balance: Correct vs. Incorrect Boundaries
3.1 The Complete Device Boundary (Correct)
For the device boundary enclosing all components (Circuit A + Circuit B + Buffer + Control), the first-law energy balance is:
\[\frac{dE_{\text{total}}}{dt} = P_{\text{in,ext}} – B_{\text{total}}(t) – P_{\text{load}}\]
where:
- \(P_{\text{in,ext}}\) — total external electrical power crossing the device boundary
- \(B_{\text{total}}(t)\) — all irreversible losses (ohmic, dielectric, radiation, discharge chemistry)
- \(P_{\text{load}}\) — useful output power to external load
- \(E_{\text{total}}\) — total stored energy inside the boundary (fields + buffer + internal storage)
This equation assumes that \(P_{\text{in,ext}}\), \(B_{\text{total}}\), and \(P_{\text{load}}\) account for all energy flows crossing the Device Boundary (including any EM radiation, acoustic, or chemical energy transport).
In steady state (\(dE_{\text{total}}/dt = 0\), time-averaged):
\[\boxed{P_{\text{in,ext}} = B_{\text{total}} + P_{\text{load}}}\]
All terms are understood as time-averaged powers over a window long compared to the regime period. For finite averaging windows, the more general form \(P_{\text{in,ext}} = \langle B_{\text{total}} \rangle + P_{\text{load}} + dE_{\text{total}}/dt\) applies; in strict steady state, \(dE_{\text{total}}/dt = 0\).
Accordingly, if \(P_{\text{in,ext}} = 0\) over a finite averaging window, then sustained nonzero \(P_{\text{load}}\) is impossible without depleting stored energy; any such depletion would appear as \(\Delta E_{\text{stored}} < 0\).
Here, \(\Delta E_{\text{stored}}\) refers to the net change of stored energy inside the Device Boundary (fields + buffer + internal storage), i.e., \(\Delta E_{\text{stored}} = E_{\text{total}}(t_2) – E_{\text{total}}(t_1)\).
The feedback power does not appear in this equation because it is entirely internal to the device boundary. It is energy being redistributed, not energy being created.
Correct efficiency:
\[\eta_{\text{true}} = \frac{P_{\text{load}}}{P_{\text{in,ext}}} \leq 1\]
3.2 The Core-Only Boundary (Source of Confusion)
If the boundary is drawn around Circuit A alone, then the feedback power from Circuit B appears as an input to the core:
\[P_{\text{in,A}}(t) = P_{\text{fb}}(t) + P_{\text{aux}}(t)\]
where \(P_{\text{aux}}(t)\) represents any external or auxiliary power delivered to Circuit A outside the feedback path (including ignition/startup as a special case; \(P_{\text{aux}}(t) = 0\) for \(t > t_0\) in steady operation).
An observer measuring only \(P_{\text{fb}}\) as “the input” and comparing it to \(P_{\text{load}}\) computes:
\[\eta_{\text{apparent}} = \frac{P_{\text{load}}}{P_{\text{fb}}} \gg 100\%\]
This is not a physics violation — it is a boundary error. The observer has:
- Counted only the maintenance channel as “input”
- Ignored that \(P_{\text{fb}}\) itself comes from Circuit B, which extracts from the same regime
- Ignored stored-energy changes and total system losses
3.3 Worked Example
Device boundary measurements (steady state, \(dE_{\text{stored}}/dt = 0\) time-averaged):
| Quantity | Value |
|---|---|
| Total external input \(P_{\text{in,ext}}\) | 2000 W |
| Total irreversible losses \(B_{\text{total}}\) | 1600 W |
| Useful output \(P_{\text{load}}\) | 400 W |
| Total stored-energy change \(dE_{\text{total}}/dt\) | 0 W |
Balance check:
\[2000 = 400 + 1600 + 0 \quad \checkmark\]
Correct efficiency:
\[\eta_{\text{true}} = \frac{400}{2000} = 20\%\]
Incorrect “apparent” efficiency (measuring only feedback channel):
If \(P_{\text{fb}} = 200\) W:
\[\eta_{\text{apparent}} = \frac{400}{200} = 200\% \quad \text{← boundary error, not physics violation}\]
4. Why High \(Q_{\text{eff}}\) Makes the Feedback Architecture Viable
4.1 Quality Factor and Maintenance Power
The effective quality factor \(Q_{\text{eff}}\) determines the ratio of stored energy to energy lost per cycle:
\[Q_{\text{eff}} \equiv 2\pi \, \frac{\langle E_{\text{stored}} \rangle}{\Delta E_{\text{loss per cycle}}}\]
The numerical prefactor depends on whether the definition is expressed in terms of energy or amplitude decay; in this paper, \(Q_{\text{eff}}\) is treated as an empirically identified regime parameter obtained from ring-down or bandwidth measurements at the dominant regime frequency.
For high \(Q_{\text{eff}}\): the regime retains most of its circulating energy each cycle. Only a small fraction needs to be replenished.
\(\langle P_{\text{fb}} \rangle\) represents the internal maintenance allocation required to compensate regime losses at a given operating point (including load-coupling effects captured by \(Q_{\text{eff,loaded}}\)). Therefore, depending on the operating point and coupling architecture, regimes may exist where \(\langle P_{\text{fb}} \rangle < \langle P_{\text{load}} \rangle\); this does not alter the device-boundary balance, which remains \(\langle P_{\text{in,ext}} \rangle = \langle B_{\text{total}} \rangle + \langle P_{\text{load}} \rangle + \langle dE/dt \rangle\).
Analogy: A heavy flywheel spinning at high speed (high stored energy) loses energy slowly to friction (low loss rate). A small motor can keep it spinning (maintenance), while a generator coupled to the same flywheel can extract substantial power — but only up to the point where total extraction plus friction exceeds the motor’s input.
4.2 Gas Discharge as Nonlinear Q-Control
The gas discharge in Circuit A is not an energy source — it is a controlled nonlinear element that shapes the regime:
Townsend avalanche (in the Townsend discharge regime) provides rapid conductivity switching:
\[n_e(x) = n_{e,0} \exp(\alpha x)\]
The energy for ionization comes from the circuit’s electric field, not from “air.”
Corona discharge provides:
- Nonlinear I–V characteristic enabling regime-adapted conductivity
- Pulsed structure that can synchronize with resonant modes
- Phase-sensitive interaction with the LC regime
Multi-channel discharge provides redundancy and adaptation — if one discharge channel degrades, others maintain regime stability.
The medium (air/gas) determines regime characteristics but does not supply net energy. It is a working medium, like water in a turbine — necessary for operation but not the energy source.
5. Architectural Isolation: Why Extraction Doesn’t Instantly Kill the Regime
5.1 The Problem in Classical Generators
In a classical generator, load directly creates counter-torque on the shaft (Lenz’s law). Increased load → immediate mechanical braking → immediate equilibrium.
5.2 The VENDOR Approach
In VENDOR, Lenz’s law still applies — but through a different mechanism:
- Extraction increases effective damping (reduces \(Q_L\))
- This reduces total \(Q_{\text{eff,loaded}}\), requiring more maintenance power
- But the nonlinear regime can adapt within its stability region before collapsing
- The BMS mediates this by adjusting feedback power dynamically
This is not a violation of Lenz’s law — back-action remains governed by Maxwell/Lenz; however, the externally observed load response is shaped by the buffer/control time constants and by the regime’s stability basin, allowing progressive rather than instantaneous response.
5.3 Stability Limits
Every regime has finite extraction limits. When extraction exceeds the stability margin:
- Gradual amplitude reduction (\(Q_{\text{eff,loaded}}\) drops too low)
- Transition to a lower-power operating point
- Complete regime collapse (if total losses exceed available maintenance power from external input)
This is physically expected behavior and confirms conservation-law compliance.
6. Summary: The Complete Picture
┌─────────────────────────────────────────────────────────────────┐ │ DEVICE BOUNDARY — First Law applies here │ │ │ │ P_in,ext ──► Circuit A ──(induction)──► Circuit B ──► P_load │ │ ▲ │ │ │ │ P_fb (feedback) │ │ │ └────── Buffer + BMS ◄──────┘ │ │ │ │ Internal: P_fb is power routing, not energy creation │ │ External: P_in,ext = B_total + P_load + dE/dt │ │ Efficiency: η = P_load / P_in,ext ≤ 1 │ └─────────────────────────────────────────────────────────────────┘
Key conclusions:
- The internal feedback path from Circuit B provides the maintenance power allocation for Circuit A. A fraction of extracted power is routed through the regulated DC bus to compensate core regime losses; sustained operation with nonzero \(P_{\text{load}}\) requires nonzero time-averaged \(P_{\text{in,ext}}\) at the Device Boundary.
- The buffer + BMS regulates this internal power routing, smoothing transients and protecting the nonlinear regime from destabilization.
- High effective quality factor \(Q_{\text{eff}}\) allows the regime to sustain large internal energy circulation with small maintenance power — making the feedback architecture viable.
- “η > 100%” is always a boundary or measurement-incompleteness error. When measured at the correct device boundary with complete instrumentation, the system obeys conservation laws.
- Nonlinearity modifies dynamics, not conservation. The regime-based architecture provides engineering advantages (self-stabilization, load adaptation, progressive back-action) but does not create energy.
Summary Table: Energy Flows and Common Accounting Errors
| Quantity | Meaning | Where it appears | Common mistake |
|---|---|---|---|
| \(P_{\text{in,ext}}\) | Total external input at device boundary | \(\frac{dE_{\text{total}}}{dt} = P_{\text{in,ext}} – B_{\text{total}} – P_{\text{load}}\) | Counting only \(P_{\text{fb}}\) as “input” |
| \(P_{\text{fb}}\) | Feedback power from Circuit B to Circuit A (internal) | Internal redistribution channel | Treating it as the sole input and computing η > 100% |
| \(B_{\text{total}}\) | All irreversible losses at boundary | Loss term in balance | Ignoring load-induced losses |
| \(P_{\text{load}}\) | Useful output to external load | Measured at output terminals | Assuming it is “maintained” by \(P_{\text{fb}}\) alone |
| \(dE_{\text{stored}}/dt\) | Stored-energy change (buffer + fields) | Balance equation | Ignoring buffer charge/discharge |
Appendix A: Symbols and Notation
| Symbol | Meaning | Unit |
|---|---|---|
| \(P_{\text{in,ext}}\) | Total external input power | W |
| \(P_{\text{load}}\) | Useful output power | W |
| \(P_{\text{fb}}\) | Feedback power (internal) | W |
| \(P_{\text{aux}}\) | Auxiliary/external power to Circuit A (incl. startup) | W |
| \(B_{\text{total}}\) | Total irreversible losses | W |
| \(E_{\text{total}}\) | Total stored system energy | J |
| \(\eta\) | Efficiency | — |
| \(Q_{\text{eff}}\) | Effective regime quality factor | — |
| \(\omega_0\) | Resonant angular frequency | rad/s |
| \(L\), \(C\) | Inductance, capacitance | H, F |
| \(\alpha\) | First Townsend coefficient | m⁻¹ |
| \(\sigma\) | Medium conductivity | S/m |
| \(\Phi_B\) | Magnetic flux | Wb |
| \(\mathcal{E}\) | Electromotive force | V |
Appendix B: Disclosure Note
This article presents a mathematical and boundary-correct formulation of energy balance for the VENDOR.Energy™ architecture. Its purpose is to eliminate interpretation errors — especially the “η > 100%” paradox — not to disclose full implementation details.
Numerical examples are conservative, model-level illustrations. Power capability is a function of configuration (architecture, stability margins, extraction design, thermal constraints), and implementation parameters are disclosed only through controlled documentation to qualified parties via the Silent Pitch Room.
References
- Patent documentation: WO2024209235 (PCT); ES2950176 (Spain); EUIPO No. 019220462
- Maxwell, J. C. A Treatise on Electricity and Magnetism — foundation of electromagnetic induction (Circuit B)
- Griffiths, D. J. Introduction to Electrodynamics (4th ed.) — ohmic, dielectric, and radiation losses
- Khalil, H. K. Nonlinear Systems (3rd ed.) — stability analysis and limit cycles
- Q factor: Wikipedia; Zurich Instruments
- Townsend discharge: Wikipedia
- Parametric resonance: Caldwell (2016), Clemson Thesis
- Open thermodynamic systems: Britannica
- Limit cycles: Wikipedia
- VENDOR documentation: vendor.energy
