Electrical  Characterisation

Some of the physical properties of DBDs and PBRs can be determined by analysis of their electrical characteristics. There are a number of different methods that can be used to study the behavior of the plasma discharge, such as current and voltage signals [36, 72, 90-92], Q-V plots [36, 72, 90-95], or fast fourier transforms [36] of current pulses.

The first in depth investigation into the electrical characteristics of plasma discharges was by T.

C. Manley in 1943 [92] through oscillographic studies of industrial ozone generators for water treatment. Based on these studies he observed the intermittent nature of the DBD, identifying the discharge phase and the capacitive phase of DBD plasma breakdown, as well as the discharge potential, and a method to calculate plasma power consumption based on Q-V plots.

Since this initial discovery, there has been further development of the oscillographic methods used by Manley, often alongside equivalent electrical models that represent the DBD as a modular system of more simple electrical components, typically capacitors and resistors.

Figure 20: Characteristic waveform of DBD applied voltage and current [90]

Figure 20 shows a characteristic current and voltage waveform of a DBD with the applied potential difference being a sine wave, and the resultant total current being a displacement current (approximately a sine wave) with a number of microdischarges (superimposed pulses).

The total current and applied voltage are out of phase, with the current leading the potential difference, this is indicative of a capacitive – resistive circuit [96]. If the current leads the voltage by ¼ of a cycle (I.e. 90°) this indicates an ideal capacitor with no resistance. If the voltage and current are in phase, this indicates an ideal resistor. If the current leads the voltage by anything less than 90°, but greater than 0°, this indicates combined resistive and capacitive behavior.

Figure 21: Simplest equivalent electrical circuit of a DBD

The circuit shown in Figure 21 shows the simplest equivalent circuit that can be used to model a DBD [94]. Consisting of an AC source, a capacitor (Cg) and a variable resistor (Rgap) in parallel, connected to a second capacitor in series (Cd). The dielectric layers of the reactor are treated as an ideal capacitor, represented by Cd. The gas gap, i.e. the space in which the plasma is generated, is treated as the capacitor, Cg, and the variable resistor, Rg. In a single AC cycle, as the

applied potential difference rises such that the gas gap voltage exceeds the gas breakdown voltage, i.e. Vg > Vbd, the resistance of Rg falls and charge transfer through the gas gap begins. As the applied voltage decreases, and Vg < Vbd, the resistance of R1 tends to infinity and no charge is transferred through the gas gap.

The capacitance of Cd can be approximated using Equation 11 [90]:

𝐶_!= 2𝜋𝜀_!𝜀_!𝑙 ln 𝑑 + 𝑥𝑑

Equation 11

where ε0 = 8.854 x 10^-12 F m^-1 is the permittivity of a vacuum, εq = 3.8 is the relative permativity of quartz glass. Reactor dimensions are incorporated by l (length), d (internal radius to the the dielectric material), and x (wall thickness).

Figure 20 also shows the “discharge phase”, also originally described by Manley. The discharge phase is characterised by a number of microdischarges, observed as current pulses, that typically terminate at the point where the applied potential difference to the reactor is beginning to either decrease or increase (dependent on phase) I.e. ^!"_!" = 0. These microdischarges initate at a breakdown voltage, Vbd, which will vary dependent upon the properties of the reactor (e.g. gas composition, particle size, etc).

From the voltage and current waveforms it is possible to determine:

From the applied potential waveform - Wave shape, frequency, amplitude

From the current waveform – Pulse magnitude, pulse duration, frequency, and discharge phase duration of microdischarges

Combined voltage and current – breakdown voltage, capacitance

As it is shown in Figure 20 the voltage waveform is a relatively smooth and continuous shape that is easy to obtain accurate measurement of. The current waveform, however, features very fast, dynamic pulses that are difficult to measure and resolve accurately. This problem makes numeric determination of reactor electrical characteristics a more complex challenge than it might seem. This problem is illustrated by the wide discrepancies in literature over obtained values for DBD power consumption and capacitance via a range of methods [97], however a comparative study between the different method by Ashpis et al [98] shows that a “monitor capacitor” used to calculate power consumption via a Q-V plot yields the most accurate results.

The “monitor capacitor” is a capacitor inserted in series with the DBD between the reactor and the ground, with the potential difference (Vm) measured across it. The capacitance of the monitor capacitor (Cm) is chosen to be much larger than that of the reactor (Ccell), i.e. Cm >> Ccell [97, 98].

The instantaneous charge (Qm) on the capacitor (With an assumed constant capacitance of Cm) is determined using Equation 12:

𝑄_!(𝑡) = 𝐶_!𝑉_!(𝑡)

Equation 12

where the current (Im) through the capacitor is obtained with Equation 13:

𝐼_! 𝑡 = 𝐶_!𝑑𝑉_!(𝑡) 𝑑𝑡

Equation 13

As they are in series with each other, the current through the capacitor must be equal to the current through the DBD (Icell), i.e. Im = Icell. The instantaneous power of the reactor is therefore determined using Equation 14:

𝑃 𝑡 = 𝑉_!"## 𝑡 ∙ 𝐼_!"## 𝑡 = 𝑉_!(𝑡) ∙ 𝐶_!𝑑𝑉_!(𝑡) 𝑑𝑡

Equation 14

where Vr is the potential difference across the reactor. The average power, 𝑃, over one cycle period (T) is therefore determined with Equation 15:

𝑃 = 𝑓 𝑉_!(𝑡) ∙ 𝐶_!𝑑𝑉_!(𝑡) monitor capacitor, against the instantaneous voltage across the DBD generates a hysteresis loop known as a Lissajous figure. Equation 15 shows that the area enclosed by the Lissajous figure is equal to the energy consumed per cycle, therefore when multiplied by the frequency it gives the reactor operating power.

An example Lissajous figure generated using experimental data is shown Figure 22. The diagram also shows additional, important electrical characteristic data that can be obtained from the Lissajous figure, such as the burning voltage (Ub) of the reactor, as well as charge transferred in the plasma, and capacitances of the dielectric layer (Cdiel), the gap (Cg) and the overall cell (Ccell).

Different parts of the lissajous figure correspond to different phases in the discharge cycle of the DBD. Taken from Figure 22, lines AB and CD correspond to the “discharge off” phase, when there is no plasma formation in the gap and the gradient of the line is equal to Ccell [94]. Where Ccell is composed from the gap capacitance (Cg), and the dielectric capacitance (Cdiel), applying Kirchoff’s law this is expressed by Equation 16.

1 𝐶_!"##= 1

𝐶_!"#$+ 1 𝐶_!

Equation 16

Lines BC and DA are the “active phase” of the discharge, when a plasma breakdown occurs. The gradient of these lines should, theoretically, be equal to the capacitance of the dielectric (Cdiel) for a “fully bridged gap” [99].

Figure 22: Lissajous figure generated from sampling potential difference over a monitor capacitor, and potential difference over a DBD plasma reactor. Red dots indicate sample points of experimental data, and blue lines show linear models fitted to the data using a least squares method.

The dielectric capacitance is determined entirely by the geometry of the DBD, which in the case of a typical, unpacked, coaxial DBD can be calculated using Equation 11. If the gap is not “fully bridged”, i.e. not all of the charge accumulated on the dielectric layer is transferred during the plasma discharge phase, the capacitance measured from the gradient of the lines BC and DA is less than Cd. This occurs due to some areas of the electrode not becoming saturated with microdischarges [95, 100]. Saturation of the electrodes, and consequent full charge bridging of the gap tends to happen through the application of voltages that greatly exceed the breakdown voltage [100] or through the usage of packing materials [72].

Peeters & Van De Sanden [95] have developed an alternative equivalent electrical circuit and accompanying numerical model based on the geometry of the Lissajous figure, that can be used for instances where the DBD is only partially discharging. This alternative equivalent geometry, shown in Figure 23, splits the circuit into a non-discharging (α) fraction, and a discharging fraction (β).

Figure 23: Alternative equivalent circuit of a partially discharging DBD, presented by Peeters

& van de Sanden [95]

This alternative equivalent electrical circuit and accompanying mathematical treatment, addresses the problem of the calculated (a.k.a. effective) dielectric capacitance from the Lissajous figure (ζdiel) (calculated from the gradient of lines BC and DA in Figure 22), being less than the actual dielectric capacitance of the reactor, Cdiel. However, in order to apply this alternative mathematical treatment of the DBD reactor requires the value of Cdiel to be well defined.

In a packed bed reactor, the capacitance of the packing material should also be taken into account. Mei et al [72] have proposed a simple model where the gap capacitance, Cgap, is treated as the capacitance of the gas (cg) and packing material (cp) as though they were two parallel plate capacitors. This is a simplified relationship, and although the actual capacitive behavior of the packed bed is more complex than this simplified model, it cannot be easily described by a simple mathematical relationship. In reality, each packing particle would act as an individual capacitor of unknown capacitance. However, this approach to equivalent circuit modeling would be very difficult to apply an actually packed bed DBD, hence the model proposed by Mei et al [72] is currently the best approach to mathematical representation of packed bed DBDs.