# Distribution of Magnetic Flux Density under Stress and Its Application in Nondestructive Testing
## 3 Results, Analysis, and Discussion
### Analysis Area and Comparison
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multicolumn{6}{c}{**Transformation Coefficient Values, Error, and Features**} \\ \cline{2-7}  & **1 st Coefficient** & **2 nd Coefficient** & **3 rd Coefficient** & **4 th Coefficient (\(\mathbf{y}_{\text{y}}\))** & **Error (\(\mathbf{\sqrt{E_{\text{a}}}}\))** & **Constancy (C)** & **Linearity (L)** \\ \cline{2-7}  & \((\mathbf{\
u}_{0})\) & \((\mathbf{\
u}_{1})\) & \((\mathbf{\
u}_{2})\) & **4 th Coefficient (\(\mathbf{y}_{\text{y}}\))** & **Error (\(\mathbf{\sqrt{E_{\text{a}}}}\))** & **Constancy (C)** & **Linearity (L)** \\ \hline
1 & \(-94.5\) & \(1.841\times 10^{4}\) & \(-1.082\times 10^{6}\) & \(1.968\times 10^{7}\) & \(0.0316\) & \(4.5476\times 10^{-6}\) & \(8.8671\times 10^{-4}\) \\
2 & \(2.78\) & \(-445.5\) & \(2.203\times 10^{4}\) & \(-3.416\times 10^{5}\) & \(0.0906\) & \(7.6358\times 10^{-6}\) & \(0.0012\) \\
3 & \(0.543\) & \(-89.72\) & \(4366\) & \(-6.493\times 10^{4}\) & \(0.0975\) & \(7.8258\times 10^{-6}\) & \(0.0013\) \\
4 & \(-0.4634\) & \(-362.1\) & \(1.35\times 10^{4}\) & \(-1.018\times 10^{5}\) & \(0.0300\) & \(4.0065\times 10^{-6}\) & \(0.0031\) \\
5 & \(-0.7341\) & \(47.89\) & \(-2050\) & \(2.52\times 10^{4}\) & \(0.0283\) & \(2.6892\times 10^{-5}\) & \(0.0018\) \\
6 & \(-2.642\) & \(461.2\) & \(-2.377\times 10^{4}\) & \(3.742\times 10^{5}\) & \(0.1345\) & \(6.6310\times 10^{-6}\) & \(0.0012\) \\
7 & \(33.92\) & \(-792.9\) & \(-1.8853\times 10^{4}\) & \(1.752\times 10^{5}\) & \(0.0316\) & \(1.7437\times 10^{-4}\) & \(0.0041\) \\
8 & \(-0.04409\) & \(23.83\) & \(-743.2\) & \(6368\) & \(0.0900\) & \(6.1794\times 10^{-6}\) & \(0.0034\) \\
9 & \(-0.01645\) & \(7.931\) & \(-248.5\) & \(1880\) & \(0.0980\) & \(7.6998\times 10^{-6}\) & \(0.0037\) \\
10 & \(-2.355\) & \(-55.7\) & \(22.79\) & \(1.312\times 10^{4}\) & \(0.0300\) & \(1.7843\times 10^{-4}\) & \(0.0042\) \\
11 & \(-0.4443\) & \(3.218\) & \(-104.3\) & \(1048\) & \(0.0300\) & \(3.8450\times 10^{-4}\) & \(0.0028\) \\
12 & \(0.2511\) & \(-66.29\) & \(2156\) & \(-1.86\times 10^{5}\) & \(0.1353\) & \(1.3341\times 10^{-6}\) & \(3.5231\times 10^{-4}\) \\
13 & \(-46.01\) & \(763.3\) & \(4817\) & \(-3.596\times 10^{4}\) & \(0.0316\) & \(0.0011\) & \(0.0187\) \\
14 & \(0\) & \(-2.472\) & \(-143.8\) & \(1512\) & \(0.0906\) & \(0\) & \(0.0015\) \\
15 & \(0.04189\) & \(-0.2911\) & \(-52.87\) & \(678.8\) & \(0.1015\) & \(5.7230\times 10^{-5}\) & \(3.9783\times 10^{-4}\) \\
16 & \(-1.643\) & \(-31.71\) & \(-196\) & \(4541\) & \(0.0949\) & \(3.4454\times 10^{-4}\) & \(0.0067\) \\
17 & \(-0.384\) & \(-1.047\) & \(-40.25\) & \(672.9\) & \(0.0316\) & \(5.3767\times 10^{-4}\) & \(0.0015\) \\
18 & \(-0.4819\) & \(2.362\) & \(319.4\) & \(-3229\) & \(0.1342\) & \(1.3572\times 10^{-4}\) & \(6.6556\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Values of the transformation coefficients, error, and features for each case. Moreover, by taking a closer look at the coefficients, \(p_{0}\), \(p_{1}\), \(p_{2}\), and \(p_{3}\), shown in Table 2, some features can be established; among them, the constancy **(C)** and linearity described in Equations (9) and (10), and portrayed in Figure 11:  
\[C=\frac{|p_{0}|}{|p_{1}|+|p_{2}|+|p_{3}|} \tag{9}\]  
\[L=\frac{|p_{1}|}{|p_{0}|+|p_{2}|+|p_{3}|} \tag{10}\]  
Detailed information regarding the error and the characteristics is provided in Figure 11. These data serve as a guideline for classifying cases that possess the appropriate relationship. In Figure 8a-f, which represent cases 1 to 6, the \(\Delta B_{X}\) range is small because the sensitivity of the sensor is high; for a proper assessment, these are not included in the analysis of Table 2 and Figure 11.  
As an indication of the datum quality by the given Equations (8)-(10), a data plot of these data from 18 cases is presented in Figure 11. The error quality was obtained by dividing the sum of the squared errors by the sum of the squared constraints, as indicated by Equation (8).