## 4 Results and Discussion
### Experimental Validation of Current Injection Models
#### 4.2.2 Heaviside Step Function  
Case 1: Current Flows in and out of the Same Boundary Surface  
Figure 13 illustrates the boundary conditions of Case 1 for the model using the Heaviside step function. Since current \(I\) enters and exits through electrodes of a certain width \(w\), the height of the Heaviside step function becomes \(I/w\). Therefore, the integral function containing \(f(y)\) is as follows:  
\[\int_{0}^{b}f(y)\cdot\cos\Big{(}\frac{n\pi y}{b}\Big{)}dy=\int_{y_{1}-w/2}^{y_ {1}+w/2}\frac{1}{\sigma}\cdot\frac{I}{w}\cdot\cos\Big{(}\frac{n\pi y}{b}\Big{)} dy-\int_{y_{2}-w/2}^{y_{2}+w/2}\frac{1}{\sigma}\cdot\frac{I}{w}\cdot\cos \Big{(}\frac{n\pi y}{b}\Big{)}dy \tag{34}\]  
where, \(I\) is the injected current, \(\sigma\) is the electrical conductivity of concrete, \(w\) is the electrode width, \(y_{1}\) and \(y_{2}\) are the electrode positions where current flows in and out, and \(a\) and \(b\) are the lengths of the concrete sample along the \(x\)- and \(y\)-axes, respectively. The coefficients of the series \(A_{n}\) for the electric potential field equation can be determined by deriving Equation (35) from Equation (34). Here, \(A_{n}\) of Equation (27) is calculated by Equation (35), so that \(u(x,y)\) contains the injected current flow described by the Heaviside step function. \[ \begin{split} A_{n}&=\frac{2}{n\pi\sinh\left(\frac {n\pi y}{b}\right)}\cdot\Big{(}\int_{0}^{b}f(y)\cdot\cos\big{(}\frac{n\pi y}{b} \big{)}dy\Big{)}\\ &\quad\frac{2\cdot(1/\sigma)}{n\pi\sinh\left(\frac{n\pi y}{b} \right)}\cdot\Big{(}\int_{y_{1}-w/2}^{y_{1}+w/2}\frac{1}{w}\cdot\cos\big{(} \frac{n\pi y}{b}\big{)}dy-\int_{y_{2}-w/2}^{y_{2}+w/2}\frac{1}{w}\cdot\cos \big{(}\frac{n\pi y}{b}\big{)}dy\Big{)}\end{split}  \tag{35}\]  
Table 5 shows a comparison of the measured and calculated voltages. The fitted electrical conductivity was 16.010 mS\(\cdot\)m\({}^{-1}\), and the RAE rate was 21.987%. Compared to the Dirac delta function, the Heaviside step function marginally reduced the RAE. This reduction came from V1 and V3 of Sets #1 and #2, which had high residuals. Since the results obtained from the Dirac delta and the Heaviside step functions were similar, their histograms and Q-Q plots were also similar. Case 2: Current Flows in from one Boundary and out from the Opposite Boundary  
Assuming that an equally distributed current flows in and out through electrodes of width \(w\), the configuration can be expressed as shown in Figure 14. If the current is injected into the electrode of width \(w\) at \(y=y_{1}\) on the left boundary of the rectangular area and flows to another electrode at \(y=y_{2}\) on the right boundary, \(g(y)\) and \(f(y)\) of the boundary condition can be expressed as follows:  
Figure 13: Boundary illustration of the current injection model using the Heaviside step function for Case 1 (\(w\): electrode width). \[g(y)=\left\{ \begin{array}{cc}\frac{1}{\sigma}\cdot\frac{l}{w},&y_{1}-\frac{w}{2} \leq y\leq y_{1}+\frac{w}{2}\\ 0,&\text{otherwise}\end{array} \right. \tag{36}\] \[f(y)=\left\{ \begin{array}{cc}-\frac{1}{\sigma}\cdot\frac{l}{w},&y_ {2}-\frac{w}{2}\leq y\leq y_{2}+\frac{w}{2}\\ 0,&\text{otherwise}\end{array} \right.\]  
where, \(I\) is the injected current, \(\sigma\) is the electrical conductivity of concrete, \(w\) is the electrode width, and \(y_{1}\) and \(y_{2}\) are the electrode positions where the current flows in and out, respectively. Functions \(f(y)\) and \(g(y)\) are applied to obtain \(A_{n}\) and \(B_{n}\) as follows:  
\[\left[ \begin{matrix}0&\frac{\frac{n\pi}{p}}{b}\\ \frac{n\pi}{b}\text{sinh}\big{(}\frac{n\pi a}{b}\big{)}&\frac{n\pi}{b}\text{ cosh}\big{(}\frac{n\pi a}{b}\big{)}\end{matrix} \right]\cdot\left[ \begin{matrix}A_{n} \\ B_{n}\end{matrix} \right]=\left[ \begin{matrix}-\frac{2}{b}y_{1y_{1}-\frac{w}{2} }^{y_{1}+\frac{w}{2}}\frac{1}{\sigma}\cdot\frac{l}{w}\cos\big{(}\frac{n\pi y }{b}\big{)}dy\\ \frac{2}{b}\int_{y_{2}-\frac{w}{2}}^{y_{2}+\frac{w}{2}}-\frac{1}{\sigma}\cdot \frac{l}{w}\cos\big{(}\frac{n\pi y}{b}\big{)}dy\end{matrix}