## 3 Sensitivity Optimization
### Finite Element Modelling
#### 3.1.3 Experimental Studies for Frequency Optimization  
To validate the results obtained from the FE modeling, an experimental study was carried out on mild steel S275 JR plates. An AD 5933 evaluation board was used for complex impedance measurements, which uses the commercially available high precision impedance analyzer chip [33]. The AD 5933 has a built-in 16 MHz clock and a sampling rate of 1.04 MHz It has an onboard digital signal processing (DSP) engine to carry out a discreet Fourier transform (DFT). The DFT algorithm returns a real and imaginary value for the impedance calculation. The board can measure the impedance from 100 \(\Omega\) to 1 M\(\Omega\)  
Figure 6: (**a**) The adaptive mesh used for modeling the coil, and (**b**) the convergence of coil reactance with the increasing resolution of narrow regions. Figure 7: The normalized reactance values for steel plates at different excitation frequencies. and carry out a frequency sweep from 1 kHz to 100 kHz. A higher frequency can also be measured with a small error. The measurement procedure requires a simple inverting amplifier (resistor of known impedance) for calibrating the system. The inverting amplifier (\(R_{fB}/Z_{unknown}\)) senses the current in the unknown impedance and converts it into voltage [34,35]. The _Gain factor_ is calculated by the formula:  
\[Gain\ factor=\frac{\left\lfloor\frac{1}{Impedance}\right\rfloor}{Magnitude} \tag{11}\]  
When the device under test (DUT) is connected to the board, the measurements are multiplied by the gain factor evaluated in the calibration step and then impedance is calculated. Both imaginary and real parts of impedance, along with phase, are returned. The sensor was tested on plates with thicknesses of 4, 5, 6, 7, and 8 mm. The normalized reactance was calculated by subtracting the reactance of all the plates from the 8 mm thick plate sample. The frequency was swept from 1 kHz through to 300 kHz, with an increment of 1 kHz. The results are shown in Figure 8.  
Figure 8 shows that the reactance signals follow a similar trend as those in the FE model. A theoretical explanation of the behavior of the reactance with the frequency variation when a coil is placed above the ferromagnetic sample is given below. As evident from Figure 9, there are two sets of magnetic fields generated when an eddy current coil is placed near the ferromagnetic sample: the primary magnetic field due to the excitation current and the secondary magnetic field produced by the induced eddy currents. The primary magnetic field intensity increases with the frequency and is magnified by the permeability of the material. Meanwhile, the secondary magnetic field also increases with the increasing induced eddy currents' intensity at higher frequencies. The secondary magnetic field opposes the primary magnetic field, thus decreasing the reactance. The effect of the secondary magnetic field is not very high at lower frequencies due to the low eddy current intensity. At a certain frequency, they cancel each other out, making the reactance zero, which is called the zero-crossing frequency. Beyond this frequency, the secondary magnetic field dominates the primary magnetic field, making the reactance  
Figure 8: The normalized reactance for steel plates with different thicknesses obtained from experiment. appear negative. Additionally, as the saturation of eddy currents takes place at higher frequencies, the rate of change of reactance becomes constant. Both primary and secondary magnetic field strengths seem to depend on the permeability of the sample under test. From the above discussion, it is clear that there are two optimal frequencies in the sensitivity curve, and the choice of frequency depends on the type of defects and magnetic properties of the material under test. Though the FE model results agree with the experimental results at some points, there are a few dissimilarities: (1) the zero-crossing frequencies are shifted from 35 kHz to 54 kHz when the plate thickness is changed from 8 mm to 4 mm in the model.