Authors: O. Krishevich, V. Peretyachenko

Scope & Critical Reading Prerequisites

This article explains an analytical framework for describing self-oscillatory regimes, feedback, resonance, and synchronization in a multimodule corona-discharge generator. It is not a public performance claim, not a statement of “energy creation,” and not a substitute for independent metrology (simultaneous voltage/current waveform power measurement, uncertainty budgeting, and thermal balance verification).

In this text, the phrase “closed loop” refers to a closed feedback loop of signals and state variables that can establish a stable limit cycle (self-oscillation) by compensating internal losses with “pumping” from an explicitly defined supply and boundary conditions. It does not mean a closed thermodynamic system, and it does not imply an energy source “from air.”

Any conclusions about net energy balance, efficiency, or output power require a formally defined system boundary and validation under a documented measurement protocol. Where numeric coefficients appear below, they represent model parameters or measured transfer/gain factors inside the loop (e.g., impedance switching, resonant amplitude ratios), not a claim of net energy gain beyond total measured active input.

Introduction

The concept of a “closed loop” in a multimodule corona generator describes a self-oscillating regime with positive feedback, in which the energy supplied for start-up and sustaining operation is redistributed between high-Q resonant elements and the plasma, forming a stable limit cycle. The model does not violate thermodynamics: it relies on well-known nonlinear dynamics, negative differential resistance in a constrained regime, and a balance between losses and controlled pumping inside the defined system boundary.

Fundamental Physical Principles

Corona Discharge as the Basis of the Regime

The onset threshold of corona discharge depends on electrode geometry (often discussed using Peek-type engineering relations for corona onset in air) and on the reduced electric field E/p. In air near 1 atm, surface fields associated with corona onset can reach tens of kV/cm, varying strongly with curvature radius, surface condition, contamination, humidity, and local microgeometry.

A simplified normalized description of avalanche ionization is commonly expressed through the Townsend form:

$$ \frac{\alpha}{p} = A \cdot \exp\left(-\frac{B \cdot p}{E}\right) $$

Here \(\alpha\) is the first Townsend coefficient, \(p\) is pressure, and \(A, B\) are gas-dependent constants (with commonly cited order-of-magnitude values for air under standard conditions). The normalization emphasizes the universality of the dependence on the reduced field \(E/p\).

Key mechanism: electrons produced by background ionization are accelerated in the electric field and can ionize additional molecules on collision, producing avalanche growth in charged-particle population (Townsend avalanche). This regime provides the physical basis for a controllable, strongly nonlinear conduction element.

Nonlinear Plasma Dynamics and Negative Differential Resistance

In corona discharge, a nonlinear plasma medium is formed. Under strong fields, the electron energy distribution can deviate from Maxwellian, which changes effective transport and reaction rates and produces strongly nonlinear current–voltage behavior.

In certain operating windows, the discharge can exhibit a region of negative differential resistance (locally \(dV/dI < 0\)) in an effective circuit sense. This does not imply energy creation; it indicates that the discharge acts as an active nonlinear element within a loop, capable of sustaining oscillations by converting supplied energy into oscillatory energy while balancing losses.

Positive Feedback as the Core “Closed-Loop” Mechanism

Minimum Loop Condition

The regime becomes self-exciting when the closed-loop transfer exceeds unity in magnitude under the appropriate phase condition:

$$ K_{\text{loop}} = K_{\text{gain}} \times K_{\text{fb}} > 1 $$

where \(K_{\text{gain}}\) is the effective gain of the active nonlinear element (the discharge plus associated impedance-switching dynamics) and \(K_{\text{fb}}\) is the feedback coefficient established by the resonant network and coupling paths.

Phase balance condition: for stable oscillation, the net phase shift around the loop must satisfy \(2\pi n\) (integer \(n\)). This is the standard oscillator condition from generator theory.

Van der Pol Oscillator as a Minimal Model

The qualitative behavior can be mapped to the Van der Pol equation:

$$ \ddot{x} – \mu(1-x^2)\dot{x} + x = 0 $$

where \(\mu > 0\) sets the nonlinearity. At small amplitudes the system exhibits “negative damping” (effective pumping), while at larger amplitudes dissipation dominates, leading to a stable limit cycle (attractor). This captures the general mechanism of self-oscillation: growth from noise/perturbation to a bounded steady oscillation under nonlinear saturation.

Multimodule Architecture and Synchronization

Spectral Overlap and Stabilization

A multimodule system can exhibit spectral overlap of operating frequencies among discharge modules. If individual modules operate at slightly different frequencies with overlapping spectra, the ensemble can provide:

• Statistical stabilization: fluctuations of individual modules average out;

• Drift compensation: parameter variations in one module can be partially compensated by others;

• Synergistic coupling effects: under certain coupling strengths, partial coherence can emerge.

Electromagnetic Coupling and Kuramoto-Type Synchronization

Modules can be coupled through weak electromagnetic interaction (capacitive/inductive coupling through the surrounding dielectric and shared structures). A standard mathematical abstraction is the Kuramoto model, where the degree of phase synchronization is described by an order parameter \(r\):

$$ r e^{i\Psi} = \frac{1}{N}\sum_{j=1}^N e^{i\theta_j} $$

Here \(r \in [0,1]\) quantifies synchrony (\(r=0\) asynchrony, \(r=1\) full synchrony), and \(\Psi\) is the mean phase. In practice, experimental analogs can be extracted from spectral coherence, cross-phase maps, and time–frequency coupling measures.

Resonant Phenomena and Frequency Selectivity

Resonance Does Not Create Energy

Resonant networks redistribute supplied energy between electric and magnetic storage elements. Resonance can increase voltage or current amplitudes in specific parts of the network, but it does not create energy; total active power is determined by the defined sources, losses, and boundary conditions.

Parametric Effects

In systems where parameters of a resonant circuit are modulated, parametric amplification can occur in the standard sense (energy transfer from the modulation/pumping channel into the oscillation mode). The classic condition for parametric resonance is:

$$ \omega_{\text{mod}} = 2\omega_0 $$

where \(\omega_0\) is the natural resonance frequency and \(\omega_{\text{mod}}\) is the modulation frequency. Any such “amplification” must be interpreted as redistribution of supplied energy into a mode, not as a violation of conservation laws.

Multifrequency Resonant Structure

Nonlinear plasma systems can generate harmonics and subharmonics. A simplified representation of harmonics in a resonant structure is:

$$ \omega_n = n \times \omega_0,\quad n = 1,2,3,\ldots $$

This produces a rich spectral structure typical for nonlinear oscillatory systems with non-sinusoidal waveforms.

Energy Balance and Thermodynamic Consistency

First law: the input electrical energy (start-up and sustaining supply) is partially stored in reactive elements and plasma dynamics and partially dissipated as heat and electromagnetic radiation. The feedback loop can sustain oscillations by channeling supplied energy into the oscillatory mode while compensating losses, but it does not violate conservation.

Second law: irreversible processes (ionization, excitation, dissociation, collisions) produce entropy; the system’s total entropy production is positive. Sustained operation necessarily involves dissipative losses.

External quasi-stationary atmospheric fields, under typical ambient conditions, are not treated here as a kilowatt-scale power source. Any meaningful assessment of environmental coupling as a power channel (if ever claimed for a specific configuration) would require explicit boundary definition and conducted/radiated power-flow accounting under independent verification.

Integral “Loop Criterion” for Self-Oscillatory Feasibility

For engineering analysis, it is sometimes convenient to represent stability feasibility as a product of measurable loop-related factors (nonlinear pumping, resonance, feedback, coupling, synchrony, stabilization), under the phase-balance condition. A generic representation can be written as:

$$ K_{\text{total}} = K_1 \times K_2 \times K_3 \times K_4 \times K_5 \times \Phi_{\text{sync}} \times \Theta_{\text{stab}} $$

where each term corresponds to a measurable transfer factor (e.g., amplitude ratio on/off resonance, loop feedback factor, coupling/synchrony metrics, long-term drift stability metric). The condition for self-sustained oscillation can be expressed as:

$$ K_{\text{total}} > 1 + \delta_{\text{margin}} $$

with \(\delta_{\text{margin}}\) representing a stability margin. This is a control/oscillation criterion (signal/regime sustainment), not a statement about net energy gain beyond total active input.

Experimental Verification (Framework Statement)

Long-duration testing of multimodule corona systems can reveal complex plasma dynamics including partial synchronization, harmonic generation, and self-oscillatory modes consistent with nonlinear plasma and oscillator theory. For claims about multi-month or multi-year stability, as well as any quantified power-performance statements, independent verification in a certified laboratory with documented protocols is required.

Practical Scaling (Conceptual Form)

For a modular architecture, a conceptual scaling form (separating per-module contribution and coupling/coherence factors) can be written as:

$$ P_{\text{total}}(N) = N \times P_{\text{mod}} \times \eta_{\text{link}}(N) \times K_{\text{coh}}(N) $$

where \(\eta_{\text{link}}(N)\) represents interconnect/coupling efficiency degradation with \(N\), and \(K_{\text{coh}}(N)\) represents coherence-related enhancement/saturation effects, both calibrated from experimental data. This formula is a modeling scaffold; it does not replace metrological closure of active power and thermal balance for any конкретное устройство.

Conclusion

The “closed-loop” principle in a multimodule corona generator is scientifically grounded as a self-oscillatory regime driven by nonlinear plasma behavior, resonant redistribution, and feedback under phase-balance conditions. It is thermodynamically consistent: sustained oscillations require supplied energy and produce dissipative losses.

The framework does not claim “energy creation.” It provides a correct physical language for discussing regime formation (limit cycles), synchronization, resonance, and loop stability, and it establishes what must be measured and independently validated before any performance conclusions can be drawn.

References

1. Raizer, Y. P. Gas Discharge Physics. Springer (classic reference on gas discharge, ionization, corona/avalanche processes).

2. Lieberman, M. A., Lichtenberg, A. J. Principles of Plasma Discharges and Materials Processing. John Wiley & Sons (plasma transport, non-equilibrium EEDF, discharge fundamentals).

3. Peek, F. W. Dielectric Phenomena in High Voltage Engineering. McGraw-Hill (classic engineering reference associated with corona onset/Peek-type relations). Library of Congress record.

4. Townsend discharge and avalanche ionization (background overview). Wikipedia: Townsend discharge.

5. Van der Pol oscillator (self-oscillation, limit cycle model). Wikipedia: Van der Pol oscillator.

6. Limit cycle (nonlinear oscillator attractors). Wikipedia: Limit cycle.

7. Kuramoto model (synchronization of coupled oscillators). Wikipedia: Kuramoto model.

8. Kuramoto, Y. Chemical Oscillations, Waves, and Turbulence. Springer (foundational synchronization theory).

9. Electromagnetic coupling effects in complex plasma channels (example domain literature). Physics of Plasmas: Electromagnetic coupling effect in complex plasma channels.

10. Experimental/theoretical discussion of corona discharge oscillations and nonlinear regimes (example domain source). Corona discharge oscillations with negative differential resistance (PDF).

11. Nonlinear transition modes in plasma systems (example domain literature). Physics of Plasmas: Nonlinear study of transition modes in chaotic plasma systems.

12. Gas detector / breakdown lecture notes (Townsend, breakdown mechanisms; general educational context). Gas detector physics notes (PDF).

13. CERN technical notes on high-voltage breakdown/corona-related topics (general technical context). CERN technical report (PDF).

14. Kuramoto synchronization computational perspectives (educational context). Kuramoto synchronization (PDF).