## 2 Materials and Methods
### Partial Surface Discharging Analysis via the Electric Model  
The DBD is commonly modelled in the literature by the electric circuit shown in Figure 10. \(C_{d}\) represents the capacitive behavior of the dielectric barrier(s), in this case the series equivalent capacitance of the two quartz cylinders. The gas is modeled as a dielectric (\(C_{g}\)) as long as it is not conductive (\(\mid\)\(v_{gas}\)\(\mid\)\(<\)\(V\)th, thus \(i_{gas}\) = 0), and after breakdown it is represented with a constant voltage (\(v_{gas}\) = \(\pm\)\(V\)th), the sign of which depends on the direction of the gas current [15, 18, 19]. The parameters of this model (\(C_{d}\), \(C_{g}\) and \(V\)th) are found with experimental waveforms for voltage and current on the DBD, then tracing the electrical charge vs. voltage diagram (known as Manley's diagram) as shown in Figure 10.  
The slopes of the parallelogram are used to define the dielectric and series equivalent capacitances (respectively, \(C_{d}\) for the ON state and \(C_{eq}\) for the OFF state--\(C_{eq}\) being the value of the series connection of \(C_{d}\) and \(C_{g}\)). From the latter, the gas capacitance (\(C_{g}\)) is calculated. The zero-cross with the voltage axis determines the threshold voltage (\(V\)th). The area enclosed by the parallelogram is equal to the energy injected during one period to the DBD, so it can be used to determine the average power. It is well known that \(V\)th, \(C_{g}\), and \(C_{d}\) are physical parameters determined by the gas composition, pressure, dielectric material, and geometry; these three values define the behavior of the equivalent circuit, which remains predictive as long as the surface covered by the plasma on the electrodes, remains stable. Indeed, variations in the identified parameters as a function of the electric conditions given by the power supply are common, even if the reactor geometry does not change [11, 12]. As shown in Figure 11, this can be observed in the case of the DBD reactor used in this work: here, the electrical energy injected into the plasma, controlled by the power supply, is changed. Accordingly, the discharge covers the whole surface of the reactor ((a), \(\beta\) = 1) or only part of it ((b), \(\beta\) < 1). Note that, concerning the identified parameters of the equivalent circuit (Figure 10), \(C_{eq}\) is similar for both cases, however, identified \(C_{d}\) and \(V\)th change significantly: being physical parameters of the reactor, such changes are not acceptable. As observed in the pictures beside the [PERSON]'s diagrams, this phenomenon in the model identification takes place when the discharge occurs in a partial area of the available electrode surface (Figure 11b). Depending on the operating conditions of the power supply, the discharge area extends until it fully covers the electrodes, and the values measured using Manley's diagram coincide with the theoretical ones. In that scenario, the classic DBD electric model, shown in Figure 10, completely represents the load. Nonetheless, for the partial surface discharging case, a modified model is proposed in the literature [8, 11, 25]. In consequence, a different equivalent electric model has to be used, where \(C_{d}\), and \(C_{g}\) are both split into the non-discharging and discharging area (\(\alpha\) and \(\beta\) percentages respectively, which fulfil: \(\alpha+\beta\) = 1), as displayed in Figure 12.  
Figure 10: Simplified electric circuit model of the DBD, with current paths in ON–green and OFF–black plasma states (**a**) and its Manley’s diagram (**b**). In this case, the \(\alpha\) and \(\beta\) parameters are calculated having previously identified \(C_{d}\), \(C_{eq}\), \(C_{g}\), and \(V\)th, using the Manley's diagram for a discharge covering all the surface.