Surface dielectric barrier discharges (SDBDs) are a type of asymmetric dielectric barrier discharge (DBD) that can be used to generate ions and produce aerodynamic forces in air. They have shown promise in a range of aerospace applications, including as actuators for solid-state aircraft control or aerodynamic enhancement and as ion sources for electroaerodynamic aircraft propulsion. However, their power draw characteristics are not well understood. Whereas existing approaches use empirical functional fits to estimate the power of specific SDBD configurations, we develop here a physics-based model for SDBD power consumption that accounts for material and geometric variation between SDBDs. The model is based on models for parallel-plate or “volume” DBDs but accounts for the “virtual electrode” resulting from changing plasma length that is particular to SDBDs. We experimentally measure the power of SDBDs of three materials, eleven thicknesses, and 29 electrical operating points to find a correlation with r2=0.99 (n = 106) between model and experiment. We also use SDBD power measurements from four experiments in the literature and find a correlation with r2=0.99 (n = 101) between our model and these experiments. Since we do not use any measured parameters from those experiments in our model, this suggests that our model has the ability to robustly predict the power for different SDBD construction methods and experimental techniques. Therefore, this work provides a robust method for the quantitative design and power optimization of SDBDs for a range of engineering applications, including aerospace propulsion.

Surface dielectric barrier discharge (SDBD) actuators are non-equilibrium plasma devices capable of generating forces in air without moving parts. A typical SDBD is shown in Fig. 1, with design and operating parameters tabulated in Table I. SDBDs utilize a high-voltage (order of kilovolts) alternating current (AC) to create a temporally alternating potential difference (and hence electric field) between two electrodes.1 Under a sufficient field strength, the air is ionized, forming a non-thermal plasma, while the dielectric barrier allows a charge build-up that limits the discharge and prevents arcing between the electrodes.2 The electric field exerts a Coulomb force on the ions in the plasma that is transferred by ion-neutral collisions to the ambient air.3 The asymmetry of the electrodes results in a net body force.

FIG. 1.

Typical experimental configuration with relevant components and parameters labeled. Welec and L are generally fixed, while other parameters may vary.

FIG. 1.

Typical experimental configuration with relevant components and parameters labeled. Welec and L are generally fixed, while other parameters may vary.

Close modal
TABLE I.

Geometric and electrical operating parameters for SDBDs.

VariableDescription and units
d Dielectric thickness, mm 
f Cycle frequency, kHz 
L Electrode span, mm 
Vpp Peak-to-peak voltage, kV 
Welec Electrode width, mm 
xp Plasma extent, mm 
VariableDescription and units
d Dielectric thickness, mm 
f Cycle frequency, kHz 
L Electrode span, mm 
Vpp Peak-to-peak voltage, kV 
Welec Electrode width, mm 
xp Plasma extent, mm 

Since the 1990s, SDBDs have been considered for aerospace applications.4 They are mechanically simple and have no moving parts; they can be constructed with thin dielectric layers and thin metal electrodes (often tape or wires), making them lightweight; and they exhibit a planar, low-drag structure, making integration of SDBDs onto existing aircraft surfaces and airfoils possible without incurring added drag (for example, when retrofitting onto existing aircraft surfaces5) SDBDs are also capable of performing at relatively low power levels of tens to hundreds of watts per meter of electrode span. This is important for aircraft, where the weight of the power system can be a significant design constraint.

Aerodynamic flow control is an area in which SDBDs show promise; they can be used to manipulate the boundary layer, a thin stratum of air near the surface of an aerodynamic body that has a significant effect on its overall lifting and drag properties. In some applications, SDBDs are used to electronically trip a laminar boundary layer into turbulence in order to reduce or delay undesirable flow separation effects.6 A similar approach uses SDBDs to induce velocity immediately above the surface, injecting momentum into the boundary layer and delaying flow separation.7,8 Other research has considered SDBDs for the reduction of vortex shedding,9,10 which could enhance aircraft performance, increase mechanical longevity, and reduce noise emission. Some researchers have employed surface plasma actuators for lift enhancement and roll control, potentially eliminating the need for mechanical control surfaces on the tail and wings, which are complex, noisy, and prone to failure.11–14 

In addition to flow control, SDBDs can also be used for aircraft propulsion. Browning et al.15 and Ozturk and Jacob16 consider internal flow SDBD propulsors, while Grieg et al.17 designed a prototype SDBD thruster consisting of concentric rings. These devices use SDBDs to directly generate a force on the actuator. Xu et al.,18 on the other hand, use DBDs as an ion source for electroarodynamic (EAD) propulsion. Optimization of SDBD power consumption is important in all of these aerospace applications, in which there are slimmer performance margins and more stringent weight restrictions than ground-based systems.

SDBD power consumption is generally reported as a function of electrical, geometric, and material properties. Many studies detail the effect of peak-to-peak voltage, Vpp (Refs. 2, 8, and 19), and frequency, f (Refs. 1, 5, 6, and 20–23). The type of AC waveform may also have an effect on power consumption.19 Some authors see an effect from the dielectric permittivity and thickness,1,22,23 and others document the effect of electrode geometry.19,20

Several specific relationships have been proposed for SDBD power consumption, P; studies have proposed both quadratic relationships, i.e., PfVpp2 (Refs. 5 and 21–24), and power law scalings, i.e., PfmVppn, with m ranging from 1 to 2 and n ranging from 2 to 4 (Refs. 1, 2, 8, and 20). These only consider voltage and frequency and require experimentally determined exponents and an empirically fit proportionality constant to account for the DBD construction. For example, Pons et al.22 introduce a proportionality constant of 0.065 and an offset voltage, V0, in their expression, P=0.065(VppV0)2, while Enloe et al.19 propose P=0.3Vpp3.35 for their respective SDBD configurations.

Other models use equivalent capacitance formulations to predict power. In these, the DBD is modeled as an equivalent circuit of resistive and capacitive elements whose values can be determined experimentally.25,26 These formulations work well for parallel-plate DBDs (also know as “volume” DBDs), for which the equivalent capacitive elements have constant values throughout each cycle.27,28 SDBDs, on the other hand, demonstrate time-varying discharge characteristics, which leads to nonlinear capacitance and complicates attempts at estimating an equivalent circuit.

Pipa et al.29 present one equivalent circuit model for SDBDs that uses additional circuit elements to account for the discharge expansion. This model suggests that PfLα3V2(VV0), and while the derivation of this formulation is more analytical than previous empirical studies, it relies on a lumped, measured coefficient, α, to account for geometrical and material variation between SDBD configurations. This parameter needs to be measured for each DBD design and makes a priori optimization of the physical properties of SDBDs impossible.

Some studies have sought to eliminate the need for lumped constants and more specifically determine how power scales with geometric parameters such as thickness and gap spacing,22 and with material properties such as dielectric constant.1,23 Each of these studies nonetheless still relies on empirical measurements for precise fits and validation. These existing scalings often lack dimensional correctness (i.e., the expression for power draw does not always have units of power unless the units of the proportionality constant are adjusted) and are insufficient for quantitative engineering design and optimization.

In this work, we consider a model for the power consumption for parallel-plate DBDs; this model, although generalizable, physically intuitive, and experimentally verified, does not accurately model SDBDs. We compare parallel-plate DBDs with SDBDs and identify differences in the physical discharge processes of the two that result from the changing plasma extent in SDBDs. Modifications are proposed to the parallel-plate DBD theory based on SDBD physics to derive a generalizable model for SDBD power. This model enables the a priori determination of power draw for SDBDs using a broad range of materials, geometries, and electrical parameters.

A method to calculate the power consumption of any DBD is to plot the charge, Q(t), against the voltage, V(t), in a charge–voltage cyclogram, or Lissajous plot, the area of which is the energy dissipated per cycle, Ek, i.e.,

Ek=Q(t)dV.
(1)

In sinusoidally driven parallel-plate DBDs, the cyclogram is a parallelogram, with the gradients corresponding to the effective actuator capacitance during discharge (Ceff) and prior to discharge (C0), as shown in Fig. 2. The parallel-plate DBD power can, therefore, be derived by computing the area enclosed by the cyclogram and multiplying by the frequency,30,31 i.e.,

P=fCeffV0(VppVign),
(2)

where

Vign=V01C0/Ceff.
(3)

Vign is the ignition voltage, or the minimum voltage required for the DBD to ignite, Vpp is the peak-to-peak voltage, and V0 is the voltage drop across the dielectric, which is dependent on dielectric and gas properties.30 

FIG. 2.

Charge–voltage cyclogram plot for a parallel-plate DBD. The energy dissipated per cycle is the shaded area.20 In an SDBD, the sides of the parallelogram are curved due to the changing plasma length and effective capacitance.

FIG. 2.

Charge–voltage cyclogram plot for a parallel-plate DBD. The energy dissipated per cycle is the shaded area.20 In an SDBD, the sides of the parallelogram are curved due to the changing plasma length and effective capacitance.

Close modal

This is the method used in most equivalent circuit models for volume DBDs.25–28 This method, however, does not accurately model the power consumption for SDBDs as SDBD cyclograms do not have a constant gradient during the discharge portion of the cycle [i.e., they have a time-dependent effective capacitance Ceff(t), which we see in Fig. 4(a) of our results].20,22,24,32 To account for this, a modification is proposed to the power calculation for parallel-plate DBDs in Eq. (2), making the distinction that Ceff varies in time during the surface discharge according to

Ceff(t)=ε0εrLWeff(t)d,
(4)

where ε0 is the vacuum permittivity and εr is the relative permittivity of the dielectric or “dielectric constant.” The effective electrode width, Weff(t), is the only term on the right-hand side that is time-dependent.

The change in gradient or “rounding” for surface DBDs is due to the growth of the plasma extent in time with increasing voltage; the growth creates a virtual electrode and increases the effective electrode width. This physical phenomenon has been demonstrated in numerical models, with both the time modulation of plasma sheath edges in volume DBDs33 and plasma growth in SDBDs,34 and has been observed experimentally for SDBDs.2,35 We model this effect with

Weff(t)=W0+xp(t),
(5)

where W0 represents the effective electrode width without discharge, and xp is the plasma extent. Therefore, when no plasma is present, Weff(t)=W0 and Ceff(t)=C0. This modification is consistent with the results of Kriegseis et al., who found evidence for linear relationships between voltage, plasma extent, and effective capacitance in Fig. 12(a) of Ref. 20 and Fig. 7 of Ref. 36. Orlov presents similar evidence in Fig. 11 of Ref. 37. We, therefore, introduce a constant c1 to define the proportionality between voltage and plasma length,

xp(t)=c1(V(t)Vign),
(6)

where the Vign offset ensures zero plasma length before ignition (V(t)Vign).19 

Since the time-dependent power within each cycle is not of primary interest here (the transients can be managed by power electronics), and only the time-averaged power over many cycles, we can use a time-averaged plasma length in place of the time-dependent length without loss of ability to resolve the power draw of different electrical operating points and SDBDs. Assuming a constant plasma growth with time during the discharge phase (like that observed by Enloe et al.19) we can calculate the average plasma length,

xp¯=c1(VppVign)2.
(7)

This approximation sets a fixed effective plasma length, and therefore a fixed capacitance for the cycle that idealizes the curved portion of the cyclogram with a straight line (this linearization can be visualized in Fig. 4(a), where eight measured cycles are plotted against the assumed model cyclogram). In our final expression, we also make the mathematical approximation that V0Vign, since Ceff is observed to be much greater than C0 in Eq. (3) for this and other SDBD configurations.23 Dong et al.23 also observe that ignition voltage, and therefore V0, scales linearly with thickness and is independent of material permittivity, allowing us to use

V0=c2d+c3.
(8)

As the experimental results in the next paragraph demonstrate, these approximations do not affect the accuracy of the model.

Making these changes and substituting Eqs. (4), (5), and (7) into Eq. (2), we find a final expression for SDBD power:

P=fε0εrL(W0+c1(VppV0)2)dV0(VppV0).
(9)

This expression includes the dependency of dielectric thickness and material, as well as electrical parameters of voltage and frequency. It is also consistent with the results of some empirical fits reported in literature: Pf (Ref. 6), PL (Ref. 20), and Pεr (Ref. 23). Several authors19,20 suggest that SDBD power is not dependent on either electrode's width, Welec, as long as the exposed electrode width is greater than the maximum plasma extent (which is true for all operating points considered in this paper). We do not include this variable in our model for power; however, this assumption should be confirmed as the subject of future studies. The remaining variables that are not explicitly prescribed by the DBD configuration and operating point are constants W0, c1, c2, and c3, which are experimentally estimated and are found not to vary between experimental conditions.

This final expression completes our model for SDBD power consumption, capturing the effect of dielectric material and thickness, SDBD span, AC frequency, and peak-to-peak voltage.

To validate this model, we measured the power draw of 11 SDBD configurations each at a range of electrical operating points for a total of 106 experiments (detailed more specifically in the supplementary material). The SDBD actuators used in this study (shown schematically in Fig. 1) consisted of two copper electrodes, both of width Welec= 10 mm and span L= 100 mm. There was no separation or gap between electrodes in the x direction (perpendicular to the electrode span), and the electrode thickness differed depending on construction as described below. These electrodes were separated by a dielectric material of thickness d. Both the material and the thickness of this dielectric layer were varied. By sealing off the encapsulated electrode from air, we ensured that plasma was only formed along the upper surface. In all cases, a sinusoidal AC voltage was used.

We used three dielectric materials of varying thicknesses: Kapton tape, high-density polyethylene (HDPE), and fire-resistant printed circuit board (PCB) material (FR-4). Table II shows the combinations of dielectric materials and thicknesses that were tested. Different Kapton thicknesses were achieved by hand-layering 0.089 mm pieces of tape, while the FR-4 SDBDs were made with a PCB printer. Kapton and HDPE SDBDs were made by hand and used 0.089 mm thick copper tape electrodes, while FR-4 SDBDs used 0.036 mm copper layers.

TABLE II.

DBD configurations tested.

MaterialThickness (mm)Measured dielectric constantReported dielectric constant1,38
FR-4 0.49 5.34 5.2 
 0.69   
 0.90   
 1.11   
HDPE 0.80 2.50 2.3 
 1.00   
 1.20   
Kapton 0.27 2.23 3.5 
 0.36   
 0.45   
 0.71   
MaterialThickness (mm)Measured dielectric constantReported dielectric constant1,38
FR-4 0.49 5.34 5.2 
 0.69   
 0.90   
 1.11   
HDPE 0.80 2.50 2.3 
 1.00   
 1.20   
Kapton 0.27 2.23 3.5 
 0.36   
 0.45   
 0.71   

Dielectric constants were determined independently of the databook value by measuring the capacitance of a parallel plate capacitor with known dimensions. The Kapton tape's silicone adhesive affected its dielectric constant, which was 2.23 compared to the databook value of 3.5. The measured values for the other materials were within 10% of their published values. We used the measured values in our analysis.

Figure 3 details our electrical setup and instrumentation. An Agilent 33250A waveform generator provided a sinusoidal signal to a Trek 5/80 HS 1000 V/V amplifier, which powered the SDBD. The frequency and time-series voltage across the actuator was measured using the Tektronix 6015A high-voltage probe, sampled by the Tektronix DPO2024B oscilloscope, and smoothed using the Savitzky–Golay filter39 to remove noise. The charge across the actuator was calculated by using the Tektronix P2200 probe to quantify the time-series voltage, Vcap, across a probe capacitor with known capacitance, Ccap, connected in series between the encapsulated electrode and ground. Charge was calculated according to

Q(t)=CcapVcap(t),
(10)

and similarly smoothed. A sample time-series waveform plot for both voltage and current is presented in Fig. 4(b). Charge was then differentiated to find current, and multiplied by voltage to determine the total power.

FIG. 3.

Electrical setup and instrumentation.

FIG. 3.

Electrical setup and instrumentation.

Close modal
FIG. 4.

Comparison of modeled power consumption and experimentally measured power. The solid fit lines have a slope of 1. (a) Charge–voltage cyclogram with modeled parameters overlaid, showing the linear approximation of the curved discharge region. (b) Temporal waveform of voltage and charge.

FIG. 4.

Comparison of modeled power consumption and experimentally measured power. The solid fit lines have a slope of 1. (a) Charge–voltage cyclogram with modeled parameters overlaid, showing the linear approximation of the curved discharge region. (b) Temporal waveform of voltage and charge.

Close modal

Measurements for each configuration were taken at a frequency of 5 kHz and at 3 to 16 unique peak-to-peak voltages ranging from 4.4 kV to 10.5 kV for a total of 106 unique experiments. Experiments were replicated up to 10 times. The supplementary material depicts a more complete breakdown of the experimental parameters. Experimental uncertainty was quantified by examining the variation in power readings between replicated experiments, and a 95% confidence interval of approximately 0.2 W/m was established.

Optical, ultraviolet, and thermal images were taken of two SDBD constructions at three operating points; these can be found in the supplementary material. Thermal heating was observed with electrode surface temperatures of up to 67 °C at a voltage of 8.6 kV.

Values for the unknown constants W0, c1, c2, and c3 were determined using least squares regression models fit to the experimental data. These values are shown in Table III. In contrast to prior experiments, where empirical constants were specific to a certain SDBD configuration, these constants were found not to change across different experimental configurations or research groups. W0 can be interpreted as an effective electrode width without discharge for SDBDs with zero gap between exposed and encapsulated electrodes; c1 represents the voltage required to sustain a unit length of plasma in air; and c2 and c3 capture the relationship between dielectric thickness and V0.

TABLE III.

Value of constants (determined by experimental data fitting).

ParameterValueUnits
W0 0.89 mm 
c1 1.24 kV–1 mm 
c2 2.66 kV mm−1 
c3 2.96 kV 
ParameterValueUnits
W0 0.89 mm 
c1 1.24 kV–1 mm 
c2 2.66 kV mm−1 
c3 2.96 kV 

In order to verify the physical basis of this model, the intermediate parameters C0, Ceff, and V0 were used to reconstruct charge–voltage diagrams that were then superimposed onto the collected data. An example of this is shown in Fig. 4(a), in which eight cycles are plotted. The effect of using the average plasma length to linearize the curved discharge region can be seen. Due in part to this approximation, there was a 3% underestimation of energy per cycle. Since this error occurs during the discharge portion of the cycle, greater discrepancy would be expected during longer discharge phases, which would occur at higher peak-to-peak voltages and in configurations with lower ignition voltages than what were tested here. Quantifying the error for both this study and the selected external studies [shown in Fig. 5(b)] shows a 5.6% mean (2.1% median) underprediction.

FIG. 5.

Comparison of modeled power consumption and experimentally measured power. The solid fit lines have a slope of 1. (a) DBD configurations examined in this study. (b) DBD configurations from this and other studies.

FIG. 5.

Comparison of modeled power consumption and experimentally measured power. The solid fit lines have a slope of 1. (a) DBD configurations examined in this study. (b) DBD configurations from this and other studies.

Close modal

Power was calculated for each operating point and each DBD configuration according to our model (using the measured constants) and plotted against the corresponding experimentally measured power draw. Figure 5(a) shows the resulting comparison of these 106 points. Figure 5(b) shows our experimental data alongside 101 data points extracted from four other studies, evaluating the performance of our model using data collected by other authors.2,3,20,40 For the 101 external data points, the coefficient of determination (r2) is 0.968. For all 207 points, the coefficient of determination is 0.976, the mean absolute percent error (MAPE) is 11.5%, the median absolute percent error (MdAPE) is 6.4%, and the root mean square error (RMSE) is 0.56 W/m.

The frequencies and voltages used by these four external studies encompass a greater range than this study: 3 kHz to 14 kHz (external) vs 5 kHz (this study) and 1.8 kV to 28 kV (external) vs 4 kV to 11 kV (this study). The dielectric constants of the external studies are also comparable to the range tested here (εr=25), but a wider range of thicknesses are represented (0.1 mm to 3.0 mm). Much greater thicknesses and relative permittivities will increase the non-discharging capacitance relative to Ceff and could reduce the accuracy of our approximation that V0Vign.

A model for the power consumed by asymmetric surface DBDs was developed by drawing from existing models for other types of DBDs. This model offers a physics-based, dimensionally consistent scaling for power, providing the means to estimate power consumption without having to build and test DBD configurations, in contrast to the empirical curve-fits used to date. We validated the model with power measurements of 11 unique DBD configurations taken at multiple electrical operating conditions, totaling 106 unique experiments. We also applied our model to data from experiments by four other sets of authors to assess the model's validity beyond our own results, and found that it is appropriate for a range of DBD configurations consisting of thin (0.1 mm to 3.0 mm) dielectrics of moderate (εr=25) permittivities at typical DBD operating conditions (3 kV to 14 kV and 1.8 kV to 28 kV). This encompasses many of the lightweight, low power configurations that are potentially useful in aerospace applications.

A related SDBD mechanism is the nanosecond pulsed SDBD, in which a pulsed voltage waveform drives the discharge instead of a sinusoidal AC waveform. These have the potential to exhibit quicker response times and lower energy consumption than comparable AC SDBDs.41 In these configurations, researchers have observed discharge propagation along the dielectric surface, referred to as surface ionization waves.42 These waves are analogous to the growth of the plasma extent that our equations model and the resulting charge–voltage plots for nanosecond pulsed SDBDs exhibit the same nonlinear capacitance in the discharge region.28,43 Although nanosecond-pulsed charge-voltage plots are sufficiently different to those of sinusoidal waveforms that our specific model would not apply, our method of computing power from the geometry of these plots while capturing the changing plasma extent could be used. This should be the subject of future work.

This study is an enabler for the engineering optimization of surface DBDs and provides a model that can be used for both design work and fundamental research. Further work should assess how other outcomes—such as body force or ionization rate—scale with geometric, material, and electrical parameters.

See the supplementary material for optical, ultraviolet, and thermal images of typical discharges and the table of tested configurations.

This work was funded through the Amar G. Bose Grant. The authors would also like to thank Yiou He for her support with both PCB design and instrumentation.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Supplementary Material