# Edge Effect Analysis and Edge Defect Detection of Titanium Alloy Based on Eddy Current Testing
## 2 Edge Effect Evaluation
### Factor Screening  
For the model with a coil placed above a plate [6], the normalized coil impedance can be obtained by Equation (4). \[Z=\frac{\pi\omega\mu n^{2}}{\left(l_{2}-l_{1}\right)^{2}\left(r_{2}-r_{1} \right)}\int_{0}^{\infty}\frac{1}{\alpha^{5}}l^{2}(r_{2},r_{1})\Big{(}(l_{2}- l_{1})+\alpha^{-1}\Big{\{}e^{-\alpha\left(l_{2}-l_{1}\right)}-1\Big{\}}\Big{)} \mathrm{d}\alpha \tag{4}\]  
where \(l_{1}\) and \(l_{2}\) are the distance between the coil and the lower and upper surfaces of the metal plate. \(r_{1}\) and \(r_{2}\) are the inner and outer diameter of the coil. The coefficient \(\alpha\) is determined by angular frequency \(\omega\), permeability \(\mu\), dielectric constant \(\varepsilon\), and conductivity \(\sigma\). In order to investigate the edge effect, a variety of EM sensor parameters were investigated. According to Equation (4), five parameters of the EM sensor were selected as factors, including the ratio between the inner diameter and the outer diameter of the coil (denoted by \"\(A\)\"), coil height (denoted by \"\(B\)\"), lift-off (\"\(C\)\"), excitation frequency (\"\(D\)\"), and outer diameter (\"\(E\)\"). Based on the effect sparsity principle [23], only a few of the large group of factors were active, which means only some of the coil parameters affected the edge effect significantly. It is of necessity to perform factor screening experiments to determine the set of active parameters from the factors above. A full factorial design was used for factor screening. Factorial designs have several important advantages over traditional one-factor-at-a-time experiments, including a reduced number of experiments, and possibilities to evaluate interactions among factors [24]. Because of the high computing resource usage of simulation, the number of simulations is limited; so the single replicate of \(2^{5}\) factorial design was used. The values that correspond to the high (+) and low (-) levels and the central point (0) for each factor are listed in Table 2.  
One approach to analyze the single replicate factorial design is to firstly examine the half-normal probability plot of the estimates of the effects [25]. Insignificant effects are normally distributed, with a mean of zero and tend to fall along the blue straight line as shown in Figure 5, whereas significant effects have nonzero means and hence do not lie along the straight line. The half-normal probability plot for \(D_{30\%}\) and \(D_{half}\) is shown in Figures 5 and 6. All of the effects that lie along the blue line are negligible, whereas the large effects are far from the line. It can be seen that for \(D_{30\%}\), the significant effects that emerge from this analysis are the main effects of \"\(B\)\" (coil height), \"\(C\)\" (lift-off), \"\(D\)\" (excitation frequency), and \"\(E\)\" (outer diameter), while for \(D_{half}\), the significant effects comprise the main effects of \"\(B\)\", \"\(C\)\", \"\(D\)\" and \"\(E\)\", and \"\(CE\)\" interactions. For either \(D_{30\%}\) or \(D_{half}\), \"\(A\)\" (ratio between inner and outer diameter) and all interactions involving \"\(A\)\" did not have a significant effect on edge effect, which means that factor \"\(A\)\" could be discarded from the simulation so that the experiment became a \(2^{4}\) factorial design with two replicates. The ANOVA could then be used to determine which effects are significant to edge effect in a quantitative way. On the basis of the F-test, the effect is statistically significant if its p-value in the ANOVA output is less than a significant level. Usually the significant level is 0.05. The ANOVA is summarized in Tables 3 and 4; it can be concluded that main effects \"\(B\)\", \"\(C\)\", \"\(D\)\", and \"\(E\)\" significantly affected \(D_{30\%}\), \"\(B\)\", \"\(C\)\", \"\(D\)\", \"\(E\)\", and interaction \"\(CE\)\" significantly influenced \(D_{half}\).