# The Equivalent Circuit Approach for the Electrical Diagnostics of Dielectric Barrier Discharges: The Classical Theory and Recent Developments

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## Abstract

**:**

## 1. Introduction

## 2. Basics of DBD Operation and Challenges for the Electric Diagnostics

## 3. The Classical Electrical Theory of Ozonizers

## 4. Suggestions for the Equivalent Circuit Approach

## 5. Revision of the Equivalent Circuit Approach and the Classical Electrical Theory of Ozonizers

#### 5.1. Validity of the Equivalent Circuit Approach

#### 5.2. Determination of ${C}_{cell}$ and ${C}_{d}$

#### 5.3. Discharge Current ${j}_{R}\left(t\right)$

#### 5.4. Dissipated Energy and Relevance of the Equivalent Circuit

## 6. Further Development of the Equivalent Circuit Approach and Open Questions

#### 6.1. Discharge with Tilted Electrodes and Partial Discharging

#### 6.2. Surface Discharge

#### 6.3. Packed Bed Reactor

## 7. Summary

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DBD | dielectric barrier discharge |

$QV$-plot | charge voltage characteristics |

${C}_{cell}$ | capacitance of the reactor cell without discharge |

${C}_{d}$ | the capacitance associated with dielectric barriers of the reactor cell, sometimes able to be seen as a reactor capacitance during the active discharge phase |

${\zeta}_{cell},{\zeta}_{d}$ | capacitance values obtained from the slopes of the $QV$-plot, which can coincide with ${C}_{cell}$ and ${C}_{d}$ if a parasitic capacitance ${C}_{p}$ is negligible |

${C}_{g}$ | the capacitance associated with the gas gap of the reactor cell |

$i\left(t\right)$, $V\left(t\right)$, $Q\left(t\right)$ | measurable values: external current, applied voltage, and charge |

${i}_{off}\left(t\right)$ | current measured without discharge (discharge off) |

${j}_{R}\left(t\right)$, ${U}_{g}\left(t\right)$ | equivalent circuit parameters: discharge current and gas gap voltage |

${U}_{b},{U}_{ext}$ | the values of the gas gap voltage corresponding to the ignition (breakdown) and extinguishing of the discharge |

${Q}_{max}$ | the maximal value of the measured charge |

${V}_{max}$ | the value of the applied voltage when ${Q}_{max}$ is reached, often corresponding to the voltage amplitude or the maximum of the applied voltage |

${U}_{res}$ | the value of the gas gap voltages when ${Q}_{max}$ is reached, residual voltage |

$\alpha ,\beta $ | the relative areas of the reactor cell, normalized on the whole area, which are not influenced and occupied by the discharge, respectively |

${C}_{eff}$ | the value of the $QV$-plot derivative near to ${V}_{max}$ in the case of the sinusoidal applied voltage |

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**Figure 1.**Schematic presentation of the dielectric barrier discharge cell. Reprinted from [12] with the permission of ©AIP Publishing.

**Figure 2.**Oscilloscope screen shots of the measured electrical characteristics of DBD from [7]. (

**a**) Measured voltage $V\left(t\right)$ and current waveforms $i\left(t\right)$. Vertical lines indicate moments of switching between active (discharge on) and passive (discharge off) phases of DBD. T is the discharge period. (

**b**) Charge voltage characteristics $Q\left(V\right)$ ($QV$-plot). The arrows indicate temporal development. The screen shots are reprinted from [7] with the permission of ©Electrochemical Society.

**Figure 3.**Interpretation of the classical charge-voltage characteristics of sinusoidal voltage-driven ozonizers. (

**a**) Schematic presentation of the $QV$-plot. (

**b**) Equivalent circuits corresponding to passive (plasma-off) and active (plasma on) discharge phases.

**Figure 4.**Schematic presentation of different types of the voltage waveforms (upper line) and the corresponding $QV$-plots (lower line). (

**a**) Staircase-shaped $QV$-plot measured for a sinusoidal operated DBD [19]. ©Penerbit UTM Press. (

**b**) $QV$-plot measured for bipolar pulsed operated DBD [8]. ©V.E. Zuev Institute of Atmospheric Optics SB RAS, reproduced with permission. (

**c**) $QV$-plot measured for pulsed operation in the form of damped oscillations [20] ©IOP Publishing. Reproduced with permission. All rights reserved. The figures are reproduced with the kind permission of the authors.

**Figure 5.**Simplest equivalent circuit of a DBD. Reprinted from [12] with the permission of ©AIP Publishing.

**Figure 6.**Determination of the reactor capacitances, based on experimental data from [13]. (

**a**) Examples of $QV$-plots for DBD operated by square voltage pulses. Arrows indicate values of ${Q}_{max}$ and ${V}_{max}$. Digits enumerate selected moments in the $QV$-plot with the largest amplitude of the applied voltage. (

**b**) An example of the ${Q}_{max}{V}_{max}$ plot, reproduced from [13] with the permission of ©2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. The points are the measured values of ${Q}_{max}$ and ${V}_{max}$ for DBD operated by square voltage pulses with fast (solid circles) and slow (open circles) rise times. Solid straight lines have slopes corresponding o the capacitance, and dashed lines indicate linear fit uncertainties.

**Figure 7.**Instantaneous energy dissipated in DBD excited by square voltage pulses of 9 kV in amplitude with a rise time of 20 ns (

**a**) and 70 ns (

**b**). Experimental data from [13]. Dashed lines are the total energy defined as the integral of the product $i\left(t\right)\times V\left(t\right)$. Solid lines are the discharge energy defined as the integral of the products ${j}_{R}\left(t\right)\times {U}_{g}\left(t\right)$. Grey lines correspond to the extreme values of the energy caused by uncertainties in capacitance ${C}_{cell}$ and ${C}_{d}$ displayed in Figure 6b. Enumerated moments for fast switching (a) are the same as in Figure 6a.

**Figure 8.**Schematic presentation of DBD arrangements with variable capacitance. (

**a**) Tilted electrodes, (

**b**) surface discharge, and (

**c**) packed bed reactor.

**Figure 9.**$QV$-plots measured in [14] for DBD arrangement with tilted electrodes. (

**a**) $QV$-plot for a single amplitude of the applied voltage. Dashed lines and black circles emphasize the linear parts of the plot. (

**b**) $QV$-plots for different voltage amplitudes. ©IOP Publishing. Reproduced from [14] with permission of the authors and IOP Publishing. All rights reserved.

**Figure 10.**The equivalent circuit for partial discharging (

**a**) suggested in [14] and the equivalent circuit accounting for parasitic capacitance (

**b**).

**Figure 11.**Experimental data for surface DBD [15]. (

**a**) $QV$-plot, (

**b**) applied voltage $V\left(t\right)$ (top) and charge derivative $C\left(t\right)=\raisebox{1ex}{$\mathrm{d}Q\left(t\right)$}\!\left/ \!\raisebox{-1ex}{$\mathrm{d}V\left(t\right)$}\right.$ (bottom) waveforms, and (

**c**) effective capacitance as a function of discharge expansion $\Delta x$ for different frequencies of the applied voltage, reprinted from [15] with the kind permission of authors and ©AIP Publishing. The added color inset in (c) shows $\Delta x$ schematically.

**Figure 12.**Formation scheme of the $QV$-plot for a packed bed reactor suggested in [16]. Reprinted from [16] with the kind permission of the authors, used under the terms of the Creative Commons Attribution 3.0 license, https://creativecommons.org/licenses/by/3.0/.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Pipa, A.V.; Brandenburg, R.
The Equivalent Circuit Approach for the Electrical Diagnostics of Dielectric Barrier Discharges: The Classical Theory and Recent Developments. *Atoms* **2019**, *7*, 14.
https://doi.org/10.3390/atoms7010014

**AMA Style**

Pipa AV, Brandenburg R.
The Equivalent Circuit Approach for the Electrical Diagnostics of Dielectric Barrier Discharges: The Classical Theory and Recent Developments. *Atoms*. 2019; 7(1):14.
https://doi.org/10.3390/atoms7010014

**Chicago/Turabian Style**

Pipa, Andrei V., and Ronny Brandenburg.
2019. "The Equivalent Circuit Approach for the Electrical Diagnostics of Dielectric Barrier Discharges: The Classical Theory and Recent Developments" *Atoms* 7, no. 1: 14.
https://doi.org/10.3390/atoms7010014