## 4 Results and Discussion
### Experimental Validation of Current Injection Models
#### 4.2.3 Gaussian Function
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{1}{c}{**Experimental Configuration**} & \multicolumn{1}{c}{**Measured**} & \multicolumn{1}{c}{**Calculated**} & \multicolumn{1}{c}{\
\begin{tabular}{c} **Residual** \\ **(Mea.\(-\)Cal.)** \\ \end{tabular} } & \multicolumn{1}{c}{
\begin{tabular}{c} **1 Res.\(/\)Cal.** \\ **(\%)** \\ \end{tabular} } \\ \hline Set \#1 & V1 & 0.384 & 0.948 & \(-\)0.564 & 59.494 \\ \hline  & V2 & 0.964 & 1.058 & \(-\)0.094 & 8.885 \\ \hline  & V3 & 0.964 & 0.768 & 0.196 & 25.521 \\ \hline Set \#2 & V1 & 1.269 & 2.006 & \(-\)0.737 & 36.740 \\ \hline  & V2 & 2.433 & 2.373 & 0.059 & 2.486 \\ \hline  & V3 & 2.561 & 1.826 & 0.735 & 40.252 \\ \hline Set \#3 & V1 & 2.011 & 2.774 & \(-\)0.763 & 27.505 \\ \hline  & V2 & 3.738 & 3.431 & 0.307 & 8.948 \\ \hline  & V3 & 3.127 & 2.774 & 0.353 & 12.725 \\ \hline Set \#4 & V1 & 0.872 & 1.058 & \(-\)0.186 & 17.580 \\ \hline  & V2 & 1.367 & 1.315 & 0.052 & 3.954 \\ \hline  & V3 & 1.708 & 1.058 & 0.650 & 61.437 \\ \hline \end{tabular}
\end{table} 
Table 7: Comparisons between the measured and calculated voltages using the Gaussian function for Case 1 (unit: \(10^{-3}\) Volt). Figure 18: Histograms of residuals at different \(\sigma_{s}\) values: (**a**) \(\sigma_{s}\) = 2.0, (**b**) \(\sigma_{s}\) = 2.5, (**c**) \(\sigma_{s}\) = 3.0, and (**d**) \(\sigma_{s}\) = 3.5 (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). Case 2: Current Flows in from one Boundary and out from the Opposite Boundary  
Assuming that the current in the electrode follows a normal distribution, its distribution can be illustrated as shown in Figure 19. The current flows in from the right boundary and out from the left boundary. Therefore, \(g(y)\) and \(f(y)\), which represent the right and left boundaries, respectively, can be expressed by Equation (42). \[ \begin{split} g(y)&=\left(\frac{l}{\sigma}\right) \cdot\left(\frac{1}{\sigma_{s}\sqrt{2\pi}}\right)\cdot e^{-0.5\left(\frac{l- p_{2}}{\sigma_{s}}\right)^{2}}\\ f(y)&=-\left(\frac{l}{\sigma}\right)\cdot\left( \frac{1}{\sigma_{s}\sqrt{2\pi}}\right)\cdot e^{-0.5\left(\frac{l-p_{2}}{\sigma_ {s}}\right)^{2}}\end{split}  \tag{42}\]  
where, \(l\) is the injected current, \(\sigma\) is the electrical conductivity of concrete, \(\sigma_{s}\) is a standard deviation for the Gaussian function, and \(y_{1}\) and \(y_{2}\) are the electrode positions where the current flows in and out, respectively. Using \(g(y)\) and \(f(y)\), \(A_{n}\) and \(B_{n}\) are obtained from a matrix inverse of Equation (43). \[ \begin{split}\left[\begin{matrix}0&\frac{n\pi}{b}\\ \frac{n\pi}{b}\sinh\left(\frac{n\pi a}{b}\right)&\frac{n\pi}{b}\cosh \left(\frac{n\pi a}{b}\right)\end{matrix}\right]\cdot\begin{matrix}A_{n}\\ B_{n}\end{matrix}&=\begin{bmatrix}-\frac{2}{b}\left(\frac{l}{\sigma}\right) \cdot\left(\frac{1}{\sigma_{s}\sqrt{2\pi}}\right)\int_{0}^{b}e^{-0.5\left( \frac{l-p_{2}}{\sigma_{s}}\right)^{2}}\cos\left(\frac{n\pi y}{b}\right)dy\\ \frac{2}{b}\left(\frac{l}{\sigma}\right)\cdot\left(\frac{1}{\sigma_{s}\sqrt{2 \pi}}\right)\int_{0}^{b}-e^{-0.5\left(\frac{l-p_{2}}{\sigma_{s}}\right)^{2}} \cos\left(\frac{n\pi y}{b}\right)dy\end{bmatrix}\end{split}  \tag{43}\]  
Consequently, the necessary coefficients (\(A_{n}\), \(B_{n}\), \(a_{0}\), and \(b_{0}\)) of Equation (33), are derived by Equations (44) and (45). \[ \begin{split} A_{n}&=\frac{2\cdot\frac{l}{\sigma_{s} \sqrt{2\pi}}}{\sigma_{s}\sqrt{2\pi}}\cdot\left\{\frac{1}{n\pi\tanh\left(\frac {n\pi a}{b}\right)}\int_{0}^{b}e^{-0.5\left(\frac{l-p_{1}}{\sigma_{s}}\right)^{ 2}}\cos\left(\frac{n\pi y}{b}\right)dy+\frac{-1}{n\pi\sinh\left(\frac{n\pi a} {b}\right)}\int_{0}^{b}e^{-0.5\left(\frac{l-p_{2}}{\sigma_{s}}\right)^{2}} \cos\left(\frac{n\pi y}{b}\right)dy\right\}\\ B_{n}&=-\frac{2}{n\pi}\cdot\frac{l}{\sigma_{s}\sqrt{2 \pi}}\int_{0}^{b}e^{-0.5\left(\frac{l-p_{1}}{\sigma_{s}}\right)^{2}}\cos \left(\frac{n\pi y}{b}\right)dy\end{split}  \tag{44}\]  
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