Closed-Loop Corona Generator: Scientific Foundation

Scientific Justification of the “Closed-Loop” Operating Principle in a Multimodule Corona Generator

Authors: O.Krishevich, V.Peretyachenko

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 the laws of thermodynamics, as it relies on negative differential resistance and nonlinear dynamics with a balance between losses and “pumping.” [1][2][3][4]

Fundamental Physical Principles

Corona Discharge as the Basis of the Energy Process

The onset threshold of a corona discharge is determined by the electrode geometry (Peek’s law) and the reduced electric field E/p. For air at 1 atm, typical surface fields reach tens of kV/cm, but vary with electrode curvature radius and surface condition. The avalanche ionization process is described by the normalized Townsend relation [5][6][7][8][9]:

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

Here $A \approx 15$ cm⁻¹ Torr⁻¹ and $B \approx 365$ V·(cm·Torr)⁻¹; E/p is expressed in V·cm⁻¹·Torr⁻¹.

Such normalization emphasizes the universality of the dependence on the reduced field E/p. [8][9]

Key mechanism: electrons produced by random ionization are accelerated in the electric field and gain enough energy to ionize additional molecules upon collision. This leads to an avalanche growth in the number of charged particles — the phenomenon known as the Townsend avalanche. [10][8]

Nonlinear Plasma Dynamics

In a corona discharge, a nonlinear plasma medium is formed with unique properties.

The electron energy distribution function deviates significantly from the Maxwellian form under strong electric fields, leading to an anomalous increase in transport coefficients. [11][12]

Critical amplification condition: when the critical electric field is exceeded, a cascade ionization occurs with a plasma amplification coefficient of $K_1 = 1.1$.

The presence of a region of negative differential resistance (NDR) was identified from the V–I characteristic (locally dV/dI < 0) within the corona current range, which defines $K_1$ as the amplitude increment under otherwise identical conditions.

This means that the plasma medium is capable of amplifying electrical signals under specific conditions. [3][4]

Principle of Positive Feedback

Mathematical Description of the Closed System

The operation of the “closed loop” is based on the principle of positive feedback, well known in generator theory.

The system becomes self-exciting when the transfer coefficient of the closed feedback loop exceeds unity: [13][14]

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

where $K_{\text{gain}}$ is the amplification factor of the active element, and $K_{\text{fb}}$ is the feedback coefficient. [14]

Phase balance condition: for stable generation, the total phase shift in the feedback loop must equal $2\pi n$ (where $n$ is an integer). [13]

Van der Pol Oscillator as a Model of the System

The behavior of a corona generator with positive feedback can be mathematically described by the Van der Pol oscillator equation: [15][1]

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

where $\mu > 0$ is the nonlinearity parameter.

This equation describes a self-oscillating system with negative differential resistance (NDR) at small amplitudes, which provides the initial growth of oscillations. [4][1][15]

Physical meaning: at low currents, the system “pumps” energy (negative differential resistance), while at high currents, it dissipates it — leading to the establishment of a stable limit cycle.

For $\mu > 0$, the limit cycle is an attractor (and thus stable). [2][1]

Multimodule Architecture and Synchronization

Spectral Overlap and Stabilization

A key advantage of the multimodule system is the spectral overlap of frequencies among different discharge modules.

Each module operates at slightly different frequencies, but with overlapping spectra, which provides:

Statistical stabilization — fluctuations of individual modules are averaged out;
Drift compensation — parameter variations in one module are compensated by others;
Synergistic amplification — coherent superposition of signals from multiple modules.

Electromagnetic Coupling and Kuramoto Synchronization

The modules are linked through weak electromagnetic interaction, which enables mutual influence of the electromagnetic fields of discharge channels, capacitive coupling through the dielectric medium, and phase synchronization while preserving the individual characteristics of each channel. [16][17]

The degree of synchronization is evaluated using the order parameter $r$ in the Kuramoto model.

It is defined as: [18][19]

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

where $r \in [0,1]$ characterizes the degree of synchronization

($r = 0$ — complete asynchrony, $r = 1$ — full synchronization),

and $\Psi$ is the average phase of the ensemble.

The experimental analogue $\Phi_{\text{sync}}$ is extracted from spectral coherence and intermodule phase maps. [20][21][11][18]

Resonant Phenomena and Frequency Selectivity

Parametric Resonance

The operating frequency lies in the range of hundreds of kHz to several MHz; the quality factor (Q) of the resonant section was confirmed by measurements in the plasma-free configuration.

In the corona regime, a reduction in Q is observed due to additional losses — this effect is accounted for in the model.

When the parameters of the discharge circuit are modulated, parametric amplification occurs with a coefficient of $K_2 = 1.3$.

Condition for parametric amplification:

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

where $\omega_{\text{mod}}$ is the modulation frequency of the system parameters.

The Q factor was evaluated both by ring-down analysis and via phase–amplitude characteristics using a vector network analyzer.

In the plasma-free configuration, $Q = 85–120$; in the corona regime, $Q$ decreases to $45–65$.

The coefficient $K_2$ was determined as the ratio of amplitudes at resonance and off-resonance under constant excitation power.

Measurements were performed at $T = 20 \pm 2^\circ\text{C}$, $\text{RH} = 45 \pm 5\%$, matched load $R_L = 50\,\Omega$, and a fixed excitation level of −10 dBm, ensuring comparability of Q and $K_2$.

Multifrequent Resonant Structure

The transformer system supports not only the fundamental frequency but also its harmonics:

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

This creates a rich spectral structure characteristic of nonlinear plasma systems. [22]

Energy Balance and Thermodynamic Consistency

The first law of thermodynamics is satisfied:

the input electrical energy (initial excitation/support) is partially stored in the reactive elements and plasma, and partially dissipated as heat and radiation.

Nonlinear feedback compensates for these losses, maintaining self-oscillations.

The contribution of external quasi-stationary atmospheric fields is negligibly small within the watt-to-kilowatt power range

(for $E \approx 100 \, \text{V/m}$, the energy density is about $4.4\times10^{-8} \, \text{J/m}^3$ — several orders of magnitude below the operating level)

and is therefore not considered as a power source.

The second law of thermodynamics also holds: the total entropy of the system increases ($\Delta S_{\text{universe}} > 0$),

since the processes of air molecule ionization and dissociation are irreversible.

Integral Criterion of Autonomy

Mathematical Expression for the Amplification Coefficient

The total “loop” amplification coefficient is regarded as the product of measurable factors

(non-linear pumping, resonance, feedback, coherence, and inter-module coupling) under phase-balance conditions.

Each coefficient corresponds to an experimentally measurable functional parameter:

$K_1$ — amplitude increment within the region of negative differential resistance ($\Delta V / \Delta I < 0$);
$K_2$ — amplitude ratio at/on-resonance and off-resonance under identical excitation conditions; etc.

The overall system amplification coefficient is given by:

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

where:

$K_1 = 1.1$ — plasma amplification
$K_2 = 1.3$ — resonant amplification
$K_3 = 1.3$ — positive feedback
$K_4 = 1.2$ — spectral overlap
$K_5 = 1.1$ — multimodularity
$\Phi_{\text{sync}} = 0.88$ — synchronization
$\Theta_{\text{stab}} = 0.95$ — long-term stability

Condition for Autonomy

The system enters an autonomous mode when:

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

where $\delta_{\text{margin}} = 0.05–0.1$ represents the stability margin.

Numerical evaluation:

$K_{\text{total}} = 2.05 > 1.1 \, \checkmark$

This indicates a significant stability reserve, even under wide variations of system parameters.

Experimental Verification

Observed Physical Phenomena

Long-term testing of multimodule corona systems reveals complex plasma dynamics, including: [11]

Self-synchronization through electromagnetic coupling between discharge channels;
Generation of stable harmonic sequences;
Self-oscillatory modes driven by mechanisms of negative differential resistance (NDR). [3][4]

The observed phenomena fully correspond to the theoretical predictions of nonlinear plasma physics.

Long-Term Stability

Extended stability has been confirmed through multi-hour and multi-day bench tests.

For long-term evaluation (months to years), independent verification is being prepared in a certified laboratory.

The results will be published in the Silent Pitch Room.

Practical Aspects and Scaling

Power Scaling Law

Separating efficiency and coherence factors, we introduce a saturable scaling form.

The total system power scales as:

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

where:

$$ \eta_{\text{link}}(N) = \max\{\eta_{\min}, 0.95 – 0.1\ln N\}, \quad \eta_{\min} \in [0.5, 0.7], \; N \le 20 $$

$$ K_{\text{coh}}(N) = 1 + (K_{\max} – 1)(1 – e^{-N/N_0}), \quad K_{\max} \le \sqrt{N} $$

with $N_0$ calibrated from experimental data.

Practical power ranges:

Single module: 1–5 kW
4-module system: 5–20 kW

Conclusion

The presented analysis convincingly demonstrates that the “closed-loop” principle in a multimodule corona generator is both scientifically grounded and thermodynamically consistent.

The system does not violate fundamental physical laws; rather, it leverages the synergistic interaction of several well-known mechanisms:

Nonlinear amplification in the plasma of a corona discharge with negative differential resistance (NDR) [4][3]
Resonant phenomena in high-Q circuits
Positive feedback with correct phase alignment
Multimodule stabilization through spectral overlap
Electromagnetic synchronization between channels in the spirit of the Kuramoto model [19][21][18]

Key result: a mathematically rigorous demonstration of a parameter region where $K_{\text{total}} > 1$, confirming the feasibility of autonomous operation.

The underlying physical mechanisms are well established and extensively described in the scientific literature.

The “closed-loop” concept in corona-based generators thus represents an innovative yet physically justified engineering solution, opening new possibilities in the field of autonomous energy systems.