# Detection of Broken Bars in Induction Motors Using Histogram Analysis of Current Signals
## 2 Proposed Methodology
### Sum and Difference Histograms for Texture Classification  
A histogram shows the distribution of a numerical variable contained in a dataset that measures the frequency of values within the analyzed vector. Similarly, the sum and difference histograms (SDH) method represents the frequency of values considering the addition or subtraction between a reference point \(R_{s}\) in the dataset and its neighborhood. In this work, the histograms represent the discrete joint probability functions locally calculated using only one phase of the current signal \(S(n)\) acquired from an induction motor. The SDHs are computed in a small analysis window denoted as \(W_{s}\), which counts the frequency values inside. The \(W_{s}\) window is depicted in red in Figure 5 for illustrating the range of the local data around each \(R_{s}\) point in the current vector \(S(n)\).  
The size of \(W_{s}\) helps to find the best precision by varying distances around \(R_{s}\). The histograms are calculated in \(S(n)\) over the \(W_{s}\) windows centered at \(R_{s}\), and it is displaced  
Figure 2: Data acquisition module.  
along the vector according to the set of values in \(\Delta\). Different displacements are considered to more accurately assess the local behavior of current for the BRB faults. The \(\Delta\) values with the best performance were used to evaluate the texture features.  
Figure 4: Squirrel cage rotor images: (**a**) healthy motor, (**b**) 1 BRB, (**c**) 2 BRB.  
Figure 5: Representation of the sliding window used to calculate the SDH in the signal \(S(n)\) using \(\Delta\) displacements. The displacements to the right compute \(h_{s}\) and to the left, \(h_{d}\).  
Figure 3: Single -phase current signals of an induction motor for (**a**) HLT, (**b**) 1 BRB, (**c**) 2 BRB.  
The direction of displacements in \(\Delta\) depends on the histogram to be calculated. Then, the displacements will be applied to the right to calculate the histogram of sums \(h_{s}\) and to the left for the histograms of differences \(h_{d}\), as illustrated in Figure 6. Based on the \(\Delta\) displacements, various histograms are computed according to the amplitude of the current signal defined as \(A_{c}\). The processing will be repeated until the entire vector signal \(S(n)\) is scanned.  
The SDH establishes a relation between two points in the current signal amplitude separated by a distance \((\Delta)\). Then, these points are added or subtracted to obtain the histograms \(h_{s}\) and \(h_{d}\), respectively, as defined in Equation (1). Let \(S\) be a current signal of size \(N\), and each element \(n\in[0,N-1]\) represents a signal amplitude defined at displacement \(\Delta\). The \(h_{s}\) and \(h_{d}\) histograms are defined in Equation (1).  
\[h_{s}=S(n)+S(n+\Delta)\hskip 28.452756 pth_{d}=S(n)-S(n+\Delta) \tag{1}\]  
The domains of \(h_{s}\) and \(h_{d}\) are \(G_{s}=2,...,2A_{c}\) and \(G_{d}=-A_{c}+1,...,A_{c}-1\), respectively. For \(h_{s}\), the domain \(G_{s}\) is twice the amplitude of the current, and the domain has positive values. However, for \(h_{d}\), the amplitude domain extends from negative to positive values. In the signals, there are positive and negative values. To avoid the negative currents in \(h_{d}\) and to align both domains, negative currents are adjusted to the values of \(h_{s}\). After that, \(h_{s}\) and \(h_{d}\) are normalized considering the number of samples in the histogram as \(N\) for obtaining the probability density function of the current, as shown in Equation (2). This equation requires that vectors \(S\), \(h_{s}\), and \(h_{s}\) have the same number of elements as defined in Equation (3).  
\[\hat{h_{s}}(i)=h_{s}(i)