## 2 Working Principle  
The basic principle of the magnetic eddy current technique is based on the nonlinear relationship between the magnetization field (\(H\)) and the induced magnetic flux density (\(B\)) in ferromagnetic materials such as mild carbon steel 1002, as shown in Figure 1. When a direct current (DC) magnetization field is imposed on a mild steel sample, the increase in the induced magnetic field is higher than the applied field (\(H\)) due to the presence of unpaired electrons. Initially, there is an increase in the magnetic permeability (\(\mu\)). However, beyond a certain point (a) in the curve, the magnetic induction does not increase at the same level as \(H\) because of the alignment of the unpaired electrons in the direction of the applied magnetization field. Consequently, there is a distinct decline in permeability. The relationship between \(B\) and \(H\) is given as  
\[B=\mu_{0}(H+x_{m}H)=\mu_{0}H(1+x_{m})=\mu H \tag{1}\]  
where \(x_{m}\) is the magnetic susceptibility and \(\mu\) is the magnetic permeability of the ferromagnetic material. When a steel sample is magnetized beyond the point (a) in the Figure 1, there is a leakage magnetic flux density \(\mathrm{B_{Imf}}\), whereas the magnetic flux density inside the un-corroded sample is \(B_{1}\). If there is a decrease in the wall thickness of the sample owing to corrosion, there will be an increase in the \(\mathrm{B_{Imf}}\) and the flux density inside the sample will rise from \(B_{1}\) to \(B_{2}\) correspondingly decreasing the magnetic permeability from \(\mu_{1}\) to \(\mu_{2}\). The mechanism of the magnetic eddy current principle for the detection of corrosion can be explained by the lumped parameter reluctance network. Magnetic circuits may be viewed as simplified electric circuits, in which each component provides reluctance to the flow of magnetic flux, similar to electrical resistance in an electric circuit. However, contrary to electric circuits, the magnetic flux is not contained within the magnetic path. As the flux spreads in the air gap that surrounds the circuit components, it is referred to as the fringe field. The simplified magnetic circuit model is shown in Figure 2.  
Figure 1: Magnetization and relative permeability as a function of applied field. \(F_{1}\) and \(F_{2}\) represent the magneto motive force applied by the magnets which serve as the source of magnetic flux. The magnetic flux in any component of the circuit can be calculated using  
\[\varnothing=\frac{F}{R} \tag{2}\]  
where \(R\) is the reluctance and is computed from Equation (3). \[R=\frac{l_{l}}{\mu A} \tag{3}\]  
where \(l_{l}\) is the length of the flux path, \(\mu\) is the magnetic permeability, and \(A\) is the cross-sectional area through which flux flows. The corrosion of the sample under test results in a decrease in the cross-sectional area, subsequently increasing the reluctance of the sample. As a result, the magnetic permeability of the sample is decreased. The change in permeability can be detected by monitoring the perturbations in the impedance of the harmonically excited coil placed above the magnetized sample. According to [PERSON]'s law, the analytical expressions for computing the flux in the sample and leakage magnetic flux using the equivalent magnetic circuit can be written as:  
\[\varnothing_{plate}=\frac{F_{1}+F_{2}}{2R_{m}+R_{iron}+2R_{airgap}+R_{plate}} \tag{4}\]  
\[\varnothing_{lmf}=\frac{F_{1}+F_{2}}{2R_{m}+R_{iron}+2R_{airgap}+R_{plate}+R_{ air}} \tag{5}\]  
Considering that the reluctance of air is much larger than that of a mild steel plate, the leakage magnetic flux can be ignored when calculating coil impedance because it is much smaller. When a coil with sinusoidal excitation is placed above the magnetized steel sample, the magnetization field will affect the eddy current coil signal in two ways.