## 3 Sensitivity Optimization
### Finite Element Modelling
#### 3.1.2 AC Modelling  
The AC module of the simulation software was used for carrying out sensitivity studies. The size and number of turns in the coil are the controlled variables due to the restricted yoke size. An increase in coil size will increase the distance between the magnets, subsequently decreasing the magnetization level in the test sample. Therefore, a parametric study on the excitation frequency of the coil was conducted to optimize the sensitivity.  
The Maxwell-Ampere law is used for solving the time-harmonic electromagnetic fields and is presented as:  
\[\
abla\times H=(J+\frac{\partial D}{\partial t}) \tag{10}\]  
where the left side of the equation (curl of \(H\)) is the magnetic field produced by the stationary currents (\(J\)) and the displacement current (\(\partial D/\partial t\)) due to the deformation of magnetic vortices.  
A coil, with the parameters presented in Table 3, was modeled, and the excitation frequency of the coil was swept from 1 kHz to 300 kHz, with an increment of 10 kHz.  
Figure 4: The FE model for magneto-static study. (**a**) The flux density in an 8 mm thick sample, (**b**) the increase in the flux density in a 4 mm thick plate, (**c**) the results for the flux densities in plates with different thicknesses, and (**d**) the permeability evaluated for the uniform corrosion of 1 mm–4 mm for 8 mm thick plate.  
In eddy current coil simulations, the regions where eddy currents form require a very dense meh due to the skin effect. Therefore, a very refined mesh is required in the boundaries between the coil and the steel plate perpendicular to the eddy currents.  
In order to simplify the problem, a 2D axisymmetric coil model was built, as shown in Figure 5.  
The boundaries near the eddy current coil generation sites were simulated by using an adaptive mesh. It was found that the results did not converge even when the element size was as small as 0.1 mm. However, increasing the resolution of boundary regions perpendicular to eddy currents resulted in a convergence shown in Figure 6. The mesh parameters are listed in Table 4.  
The permeability of the sample acquired from the first step was assigned to the elements in the model to simulate the plates of various thicknesses. Because of the wide range of variation in reactance at different frequencies, the reactance values computed for different plate thicknesses were normalized by using the reactance of 8 mm thick plate as reference value. The normalized inductive reactance values computed for all the excitation frequencies are illustrated in Figure 7.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Element Size Parameters** & **Values** \\ \hline Maximum Element Size & 0.5 mm \\ \hline Minimum Element Size & 0.02 \\ \hline Maximum Element growth rate & 1.1 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The mesh parameters obtained from mesh convergence study.  
Figure 5: (**a**) The 2-D axisymmetric FE model, and (**b**) the solution for coil parameters at 100 kHz frequency.  
The FE model results indicate that the relationship between the sensitivity of the sensor and the excitation frequency is not monotonic. The sensitivity increases at lower frequencies initially, but it starts to decrease and becomes zero at a certain frequency called zero-crossing frequency [31; 32]. It becomes increasingly negative at higher frequencies and then starts to decrease slightly. The choice of the excitation frequency is, therefore, a significant part of the design of MEC sensors.