## 4 Results and Discussion
### Experimental Validation of Current Injection Models
#### 4.2.1 Dirac Delta Function
The fitted electrical conductivity was 15.907 mS\(\cdot\)m\({}^{-1}\) and the RAE was 4.372%. Thus, Case 2 had a considerably low RAE than Case 1 (22.009%). In contrast to conventional medical uses, the application of EIT to cement-based materials, which have electricity insulation characteristics, requires the electric current to pass through the sample thoroughly. This might be because Case 2 had higher voltage value than Case 1, reflecting a significant difference in the electric potential field. A comparison of the calculated potential fields is presented below. Table 4 shows the measured and calculated voltages. Large errors occurred in V1 in Sets #5 to #8. These values were potential differences between two electrodes that had a current-injected electrode between them. Therefore, for voltage measurements, selecting two electrodes that have a current-injected electrode between them should be avoided because the electric field around the current-injected electrode changes so dramatically that the voltage measurement is considerably less accurate. Figure 11 shows the histogram and Q-Q plot of residuals. The residuals exhibited a well-defined normal distribution, except for two outliers toward the minus direction obtained from V1 of Sets # 6 and # 7. As previously noted, these voltages were measured from two electrodes with a current-injected electrode between them. The Q-Q plot also shows that the residuals followed the normal distribution, except for a heavy tail toward the negative values. Generally, residuals that follow a well-defined normal distribution function indicate a reliable validation between a theoretical solution and a well-controlled measurement [15]. Figure 12 shows the electric potential field obtained from Equations (31)-(33). Compared to Case 1 (Figure 9), the change in potential difference in Case 2 occurred in a wider area. A similar phenomenon was in fact observed in Sets #3 and #4 in Case 1 (Figure 9c,d). When the two electrodes where the current flowed in and out were sufficiently far apart, a change in potential difference occurred over a large area of the domain. This clear potential difference caused higher voltage values in Case 2, making it less sensitive to voltage measurements. For this reason, the RAE in Case 2 was considerably lower than that in Case 1 (Tables 3 and 4). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Experimental Configuration** & **Measured** & **Calculated** & \multicolumn{2}{c}{**Residual** (**Mea.\(-\)Cal.**)} & \multicolumn{1}{c}{**1 Res.1/Cal.**} \\ \hline Set \#1 & V1 & 17.923 & 17.896 & 0.027 & 0.151 \\ \hline  & V2 & 8.716 & 8.882 & \(-\)0.166 & 1.869 \\ \hline Set \#2 & V1 & 16.638 & 16.826 & \(-\)0.188 & 1.117 \\ \hline  & V2 & 7.981 & 7.921 & 0.060 & 0.757 \\ \hline Set \#3 & V1 & 15.412 & 15.497 & \(-\)0.085 & 0.548 \\ \hline  & V2 & 7.090 & 6.852 & 0.238 & 3.473 \\ \hline Set \#4 & V1 & 14.015 & 14.428 & \(-\)0.413 & 2.862 \\ \hline  & V2 & 6.064 & 6.079 & \(-\)0.015 & 0.247 \\ \hline Set \#5 & V1 & 6.936 & 7.401 & \(-\)0.464 & 6.269 \\ \hline  & V2 & 13.440 & 13.298 & 0.143 & 1.075 \\ \hline Set \#6 & V1 & 4.392 & 5.558 & \(-\)1.167 & 20.997 \\ \hline  & V2 & 12.895 & 12.337 & 0.558 & 4.523 \\ \hline Set \#7 & V1 & 1.386 & 3.160 & \(-\)1.774 & 56.139 \\ \hline  & V2 & 12.166 & 11.268 & 0.898 & 7.969 \\ \hline Set \#8 & V1 & 1.332 & 1.130 & 0.202 & 17.876 \\ \hline  & V2 & 11.050 & 10.495 & 0.555 & 5.288 \\ \hline \end{tabular}
\end{table} 
Table 4: Comparisons between the measured and calculated voltages using the Dirac delta function for Case 2 (unit: \(10^{-3}\) Volt). Figure 11: Comparison between a normal distribution and residuals obtained from the Dirac delta function for Case 2: (**a**) histogram of residuals (the label of x axe: residuals (differences) between calculations and measurements in \(10^{-3}\) volt unit, the label of y axe: the number of residuals in frequency) (**b**) Q-Q plot.