## 3 Signal Analysis
### Bispectrum Analysis
#### 3.2.3. Bispectrum Features of Arc Faults  
The contours, amplitudes, phase angles and the \(1\
icefrac{{1}}{{2}}\)-dimensional slices of different load bispectra are plotted as shown in Figures 4-6. According to the differences between the normal state and arc fault bispectra, it is demonstrated that more information about the nonlinear dynamic features of the load signals is given by the bispectra. When series arc faults occur in circuits, the energy distribution which can be discovered from the contours becomes diffused. Since the power spectrum is phase blind, it is unable to detect the presence of phase coupling. However, the bispectrum gives an indication of the correlation between the phases of different frequency components \(\omega_{1}\) and \(\omega_{2}\). In addition, the power spectrum also applies the Fourier transform to all signals and is sensitive to Gaussian noise. In contrast, the bispectrum can eliminate the additive Gaussian noise and provide information of the phase coupling. The phases of electrical load arc faults are different from those of normal states, where the former increase suddenly and randomly in some frequency domain while the latter change smoothly in most frequency domain. Since the capacitive and the inductive reactance of fault arc are unstable, they may change randomly. As shown in the analysis of arc fault contours (Figures 4-6) and the symmetry features of the bispectrum, all the valuable information in the arc fault signal is contained in the bispectra in the inner triangular domain \(\omega_{2}\geq 0\), \(\omega_{1}\geq\omega_{2}\) and \(\omega_{1}+\omega_{2}\leq\pi\). Therefore, the spectrum is sufficient for the description of all bispectrum features and the symmetry features are given by:  
\[\begin{array}{l}S_{3_{u}}(\omega_{2},\omega_{2})=S_{3_{u}}(\omega_{2},\omega )=S_{3_{u}}^{*}(-\omega_{1},-\omega_{2}  
is similar, and proves they are effective in reflecting the information of the dominant frequency components of the non-Gaussian nonlinear arc fault signals. When the loads work in normal, the background noise and working processes are treated as stationary random processes, and there are few peaks in the bispectrum graphs. When series arc faults occur in circuits, as a result of the coupling relations of different high frequency components generated by arc faults, bispectrum peaks gradually change from concentrated distribution to dispersed and become mixed and disordered. The number of bispectrum peaks is evidently increased. There are increased numbers of high peaks in the arc fault bispectrum as a result of strong correlation in some frequency. The arc fault becomes more serious and the nonlinearity becomes stronger. The above dynamic features of load states are also performed in the slices of bispectra. Furthermore, the \(1\
icefrac{{1}}{{2}}\)-dimensional slices of bispectra are easier to calculate than bispectra. The phase coupling frequency components can also be extracted from the Gaussian noise environment. Bispectrum peak numbers of arc faults increase obviously, and the highest bispectrum peak positions of arc faults move towards \(\omega=0.5\pi\). Therefore, the attended phase coupling components and their generated components in the \(1\
icefrac{{1}}{{2}}\)-dimensional slices of bispectra can be used for arc fault detection. However, it is hard to find constant thresholds to classify normal states and arc faults in all loads. Thus, a good classification algorithm is needed. Figure 4: _Cont._  
Figure 4: AR bispectra of halogen lamp. (**a**) Normal contour; (**b**) Normal amplitude; (**c**) Normal phase angle; (**d**) Normal 1\({}^{\prime}\)2-D slices; (**e**) Arc fault contour; (**f**) Arc fault amplitude; (**g**) Arc fault phase angle; (**h**) Arc fault 1\({}^{\prime}\)2-D slices. Figure 5: _Cont._  
Figure 5: AR bispectra of vacuum cleaner.