# Further Investigations into the Capacitive Imaging Technique Using a Multi-Electrode Sensor
## 3 Simulation Analysis of the Multi-Electrode Sensor for Defect Detection  
### FEM Models and the Electric Potential Lines  
In this work, finite element method (FEM) models have been constructed to calculate the electric fields from the multi-electrode sensor in the process of inspection and to validate the feasibility of detecting defects for the non-conducting specimen and conducting specimen covered by an insulation layer. FEM models can be used to predict the electric fields between the driving electrode and the different sensing electrodes and predict the electrode fields between the sensor surface and different  
Figure 2: Detection principle of the multi-electrode sensor: (**a**) the multi-electrode sensor inspecting the non-conducting specimen; (**b**) the multi-electrode sensor inspecting the conducting specimen covered by an insulation layer. specimens. Because the fringing electric fields from the multi-electrode sensor are considered as quasi-static electric fields, simplified versions of [PERSON]'s equations [19] can be used, namely  
\[\
abla\times E\approx 0 \tag{1}\]  
\[\
abla\times H\approx 0 \tag{2}\]  
\[\
abla\cdot D\approx\rho \tag{3}\]  
\[\
abla\cdot B\approx 0 \tag{4}\]  
where, \(E\) denotes the electric field, \(H\) denotes the magnetic field strength or auxiliary field, \(D\) denotes the electric displacement field, \(\rho\) denotes the free electric charge density and \(B\) denotes the magnetic field. The analysis of the multi-electrode sensor can be considered as electrostatic analyses with the simplified Equations (1)-(4) and the electric fields can be simplified to be produced by various charge distributions from the multi-electrode sensor. According to Equations (5) and (6), the potential distribution \(\varphi\) can be obtained by using FEM models [11; 12]. \[E=-\
abla\varphi \tag{5}\]  
\[\
abla(\varepsilon_{O}\varepsilon_{r}\
abla\varphi)=0 \tag{6}\]  
where, \(E\) again denotes the electric field, \(\varphi\) denotes the electric potential distribution, \(\varepsilon_{0}\) denotes the electric constant or the permittivity of a vacuum, and \(\varepsilon_{r}\) denotes the relative permittivity. It is well known that electric field lines are always perpendicular to electric potential lines [20]. If a defect exists in the effective inspection range of the fringing electric field, it can cause distortion of the electric field lines and cause distortion of the electric potential lines correspondingly. The distribution of the fringing electric field lines near the defect is extremely complex. It is very difficult to obtain the distortion of the electric field lines, while it is relatively easy to obtain the distortion of the electric potential lines [21]. According to Equations (1)-(6), the electric potential lines can be acquired using FEM models. In order to clearly display this distortion, electric potential lines were employed in our FEM models. Theoretical FEM models were constructed by using the AC/DC module in COMSOL, which can be used to model the distribution of the electrical potential lines near a defect for the multi-electrode sensor. As shown in Figure 3a, a multi-electrode sensor consists of four co-planar electrodes, one being the driving electrode (No. 1) and the other three being the sensing electrodes (No. 2-4). The center distance between any two adjacent electrodes is 13 mm. Besides that, the multi-electrode sensor has a surrounding shielding electrode (No. 5) which was used to shield external stray capacitance. The total size of the multi-electrode sensor is 55 mm\(\times\)30 mm and the four co-planar electrodes have the same width (b = 10 mm) and height (h = 23 mm). To save computation cost, 2D FE models which describes the cross-section of the above mentioned sensor along the symmetric axis (as indicated by the red line in Figure 3a) and the specimen under test were used hereafter. The cross-section of the multi-electrode sensor is shown in Figure 3b. CL12, CL13, CL14 were the center lines between the driving electrode (No.