## 4 Results and Discussion
### Experimental Validation of Current Injection Models
#### 4.2.1 Dirac Delta Function
The histogram shown in Figure 8a was similar to a normal distribution, except for its tail bounds induced from V1 and V3 in the corner areas. Nevertheless, the Q-Q plot in Figure 8b shows that the residuals closely followed the normal distribution. Electric potential maps were created based on the calculations from Equations (26) and (27). As shown in Figure 9, when the current-injected point was located at an inner area, the potential distribution was concentrated at that point, whereas when the point was in a corner area, the potential was distributed over a wide area. Furthermore, a significant variation in the potential field mostly occurred near the electrodes where the current flowed in and out. For other boundary interfaces, the potential change was gradual, and the potential variation was substantially smaller at the boundary interface where the voltages were measured. This might be the reason for the high RAE (22.009%) as the sensitivity to the measurement increased. Case 2: Current Flows in from one Boundary and out from the Opposite Boundary  
As shown in Figure 10, the current was injected at \(y=y_{1}\) on the left boundary (\(x=0\)) of the rectangular area and flowed out at \(y=y_{2}\) on the right boundary (\(x=a\)). Using the Dirac delta function, the functions of boundary conditions \(f(y)\) and \(g(y)\) can be expressed as follows:  
\[g(y)=\frac{I}{\sigma}\cdot\delta(y-y_{1})\text{ and }f(y)=-\frac{I}{\sigma} \cdot\delta(y-y_{2}) \tag{29}\]  
where, \(\delta\) is the Dirac delta function, \(I\) is the current, and \(\sigma\) is the electrical conductivity. Figure 8: Comparison between a normal distribution and residuals obtained from the Dirac delta function for Case 1: (**a**) histogram of residuals and (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. Figure 10: Boundary illustration of the current injection model using the Dirac delta function for Case 2. Figure 9: Electric potential maps created based on calculation from the Dirac delta function for Case 1: (**a**) Set #1, (**b**) Set #2, (**c**) Set #3, and (**d**) Set #4 (\(x\) and \(y\) axes indicate 50 \(\times\) 50 mm corresponding to the size of the sample; the red and blue colors mean positive and negative voltage, respectively). Using Equation (29), Equation (21) is converted to the following equation:  
\[ \begin{bmatrix}0&\frac{n\pi}{b}\\ \frac{n\pi}{b}\text{sinh}\big{(}\frac{n\pi a}{b}\big{)}&\frac{n\pi}{b}\cosh\big{(} \frac{n\pi a}{b}\big{)}\end{bmatrix} \cdot \begin{bmatrix}A_{n}\\ B_{n}\end{bmatrix} = \begin{bmatrix}-\frac{2}{b}\left(\frac{l}{\sigma}\right) \cos\big{(}\frac{n\pi y_{1}}{b}\big{)}\\ \frac{2}{b}\left(-\frac{l}{\sigma}\right)\cos\big{(}\frac{n\pi y_{2}}{b}\big{)} \end{bmatrix}  \tag{30}\]  
By going through the inverse matrix process, \(A_{n}\) and \(B_{n}\) in Equation (30) can be expressed as follows:  
\[ \begin{array}{l}A_{n}=\frac{2\left(\frac{l}{\sigma}\right)}{n\pi\text{tanh }\big{(}\frac{n\pi y_{1}}{b}\big{)}}\cos\big{(}\frac{n\pi y_{1}}{b}\big{)}+ \frac{2\left(-\frac{l}{\sigma}\right)}{n\pi\text{sinh}\big{(}\frac{n\pi y_{2} }{b}\big{)}}\cos\big{(}\frac{n\pi y_{2}}{b}\big{)}\\ B_{n}=-\frac{2\left(\frac{l}{\sigma}\right)}{n\pi}\cos\big{(}\frac{n\pi y_{1} }{b}\big{)}\end{array}  \tag{31}\]  
The part, \(a_{0}\cdot x+b_{0}\), of Equation (22) is solved as follows:  
\[ \begin{array}{l}a_{0}=\frac{1}{b}\int_{0}^{b}-f(y)dy=-\frac{1}{b}\cdot\left( \frac{l}{\sigma}\right)\int_{0}^{b}\delta(y-y_{1})dy=-\frac{1}{b}\cdot\left( \frac{l}{\sigma}\right)\\ b_{0}=-\frac{a_{0}}{2}\end{array}  \tag{32}\]  
The completed series solution of \(u(x,y)\) is as follows:  
\[u(x,y)=\sum_{n=1}^{50}\Big{[}\Big{\{}A_{n}\cosh\Big{(}\frac{n\pi}{b}x\Big{)}+ B_{n}\text{sinh}\Big{(}\frac{n\pi}{b}x\Big{)}\Big{\}}\cos\Big{(}\frac{n\pi}{b}y \Big{)}\Big{]}+a_{0}x+b_{0} \tag{33}\]  
Equation (33) was fitted to the measured voltages.