# Current Filaments in Asymmetric Surface Dielectric Barrier Discharge
## 5 Experimental Results
### Rogowski Coil
#### 5.1.5 Correlations between BS  
A question that often arises in DBD is whether bumps (here called strokes) are correlated to each other [26]. The same quest applies to our SDBD results. We therefore calculated, for our records, the cross-correlations, \(C_{BS,BS}^{(i,j)}\), between BS separated by a time lag \(\Delta t=|i-j|T\), where \(T\) is the applied signal period, according to the standard definition,  
\[C_{BS,BS}^{(i,j)}=\frac{\left\langle|I_{BS}^{(i)}(t)-\left\langle I_{BS}^{(i) }\right\rangle|I_{BS}^{(j)}(t)-\left\langle I_{BS}^{(j)}\right\rangle|\right\rangle _{\Delta t}}{\sigma_{BS}^{(i)}\sigma_{BS}^{(j)}}, \tag{4}\]  
where \(I_{BS}^{(i)}(t)(>0)\) is the value of the burst current within the \(i\)th BS at time \(t\), \(\left\langle I_{BS}^{(i)}\right\rangle\) the mean current value, and \(\sigma_{BS}^{(i)}\) the corresponding standard deviation. A typical cross-correlation analysis is illustrated in Figure 17, for HV = 5.48 kV. Similar results (not shown here) are obtained for the other applied voltages. As one can see from this analysis, there is a residual or almost no apparent temporal correlations between BS. Therefore, in order to uncover a possible hidden structure of BS, we compared two consecutive BS with each other (i.e., 1-2, 3-4, etc. ), by building their burst current intensity differences as a function of time,  
\[\Delta I_{BS,BS}^{(i+1,i)}(t)=I_{BS}^{(i+1)}(t)-I_{BS}^{(i)}(t), \tag{5}\]  
for \(i=1,3,5,\ldots,2N_{p}-1\), where \(N_{p}\) is the total number of consecutive BS pairs in the record. An example is shown in Figure 18a, for \(i=1\) and HV = 5.48 kV. As one can see, bursts do not occur at the same time (note that the time origin, the start of the BS half-cycle, is the same for all strokes). We found that the time delay between bursts follows a uniform random distribution (results not shown). Next, to find the mean behaviour of BS, we average the differences in Equation (5), determined at fixed times \(t=n\Delta\), with \(1\leq n\leq N\) and \(\Delta t=0.25\) ns, over all pairs \(N_{p}\),  
\[\left\langle\Delta I_{BS,BS}^{(i+1,i)}(t)\right\rangle_{i}=\frac{1}{N_{p}} \sum_{j=1}^{N_{p}}\Delta I_{BS,BS}^{(2i,2j-1)}(t). \tag{6}\]  
Figure 16: Same as in Figure 13 for the BS currents. Note that the value HV \(\simeq\) 5.5 kV seems to play the role of a separation voltage between two regimes, as found for the charge. The mean differences are plotted in Figure 18b, where one can see that they do not average out to zero. To better visualize the evolution of differences, it is convenient to add them sequentially, as follows  
\[S(n)=\sum_{j=1}^{n}\big{\langle}\Delta I_{BS,BS}^{(i+1,j)}(j\Delta t)\big{\rangle} _{i}. \tag{7}\]  
Their time behavior can be conveniently visualized by adding the differences sequentially as for random walks (red line in the figure). The corresponding random walk (RW) profiles, for selected applied HV, are displayed in Figure 19a. In order to determine the degree of auto-correlation in the signal, in this case referred to a sequence of two consecutive BS, we rely on the calculation of the Hurst exponent (see e.g., [26; 32; 33]). To this end, we evaluate the \(n\)th (\(1\leq m\leq N/\ell\)) mean RW profile at 'time scale' \(\ell\) (\(\ell\equiv t/\Delta t\)), with \(1\leq\ell\leq N\), given by  
\[B_{m}(\ell)=\frac{1}{\ell}\sum_{n=1}^{\ell}S[(m-1)\ell+n]. \tag{8}\]  
Next, we study the fluctuations between neighboring boxes according to,  
\[F(\ell)=\big{\langle}[B_{m+1}(\ell)-B_{m}(\ell)]^{2}\big{\rangle}^{1/2}, \tag{9}\]  
which is expected to display a power-law behaviour,  
\[F(\ell)\simeq\ell^{H}, \tag{10}\]  
where \(0\leq H\leq 1\). It should be noted that fully uncorrelated RW has \(H=1/2\), and \(H=1\) for fully correlated ones. The results shown in Figure 19b suggest that initially, within about 6 ns, the neighboring BS remain strongly correlated. A second regime develops at intermediate times, \(6<t<200\) ns, suggesting that weaker correlations persist and are well described by an exponent \(H\simeq 2/3\).