## 2 Theoretical Background  
### Eddy-Current Field Equations  
Eddy-current electromagnetic analysis can be modeled using magnetic vector potential (\(\overset{\rightarrow}{A}\)) and electric scalar potential (\(\Phi\)), and applied using finite element numerical analysis. Two governing equations are used to interpret electromagnetic fields in the detection of eddy currents: [PERSON]'s law, which shows the relationship between a magnetic field and the current around the wire in which the current is flowing; and [PERSON]'s law, which states that time-varying magnetic fields produce an induced voltage (electromotive force) in a closed circuit and induce current in a wire. [PERSON]'s law can express the relationship between the intensity vector (\(\overset{\rightarrow}{H}\)) of a magnetic field and the current density vector (\(\overset{\rightarrow}{J}\)) as shown in Equation (1), and [PERSON]'s law can be represented by the magnetic density vector (\(\overset{\rightarrow}{B}\)) of an electromagnetic field vector (\(\overset{\rightarrow}{E}\)), as shown in Equation (2). In addition, magnetic flux density vectors (\(\overset{\rightarrow}{B}\)) and electric flux density vectors (\(\overset{\rightarrow}{D}\)) are shown in Equations (3) and (4) according to [PERSON]'s law [9]. \[\
abla\times\overset{\rightarrow}{H}=\overset{\rightarrow}{J} \tag{1}\]  
\[\
abla\times\overset{\rightarrow}{E}=-\frac{\partial\overset{\rightarrow}{B }}{\partial t} \tag{2}\]  
\[\
abla\times\overset{\rightarrow}{B}=0 \tag{3}\]  
\[\
abla\times\overset{\rightarrow}{D}=\rho \tag{4}\]The above Equation (1) through (4) include the equations for the composition of (a) \(\overset{\rightarrow}{B}=\mu\overset{\rightarrow}{H}\), (b) \(\overset{\rightarrow}{J}=\sigma\overset{\rightarrow}{E}\), and (c) \(\overset{\rightarrow}{D}=\varepsilon\overset{\rightarrow}{H}\), where \(\mu\) represents the self-investment rate, \(\sigma\) indicates the electrical conductivity, and \(\varepsilon\) indicates the inheritance rate. If formulas (a) and (b) are substituted for Equation (1), the following is the result:  
\[\frac{1}{\mu}\
abla\times\overset{\rightarrow}{B}=\sigma\overset{ \rightarrow}{E}. \tag{5}\]  
The magnetic induction field (\(\overset{\rightarrow}{B}\)) can be expressed as the curl of the magnetic vector potential (\(\overset{\rightarrow}{A}\)). \[\
abla\times\overset{\rightarrow}{E}=-\frac{\partial}{\partial t}\left( \
abla\times\overset{\rightarrow}{A}\right)=\
abla\times-\frac{\partial \overset{\rightarrow}{A}}{\partial t} \tag{6}\]  
\[\
abla\times\overset{\rightarrow}{E}=-\frac{\partial\overset{\rightarrow}{B }}{\partial t} \tag{7}\]  
\[\overset{\rightarrow}{E}=-\frac{\partial\overset{\rightarrow}{A}}{\partial t }-\
abla\
abla \tag{8}\]  
V represents the electrical scalar potential, and Equation (8), in relation to Equation (5), can be expressed as:  
\[\frac{1}{\mu}\
abla\left(\
abla\times\overset{\rightarrow}{A}\right)=-\sigma \frac{\partial\overset{\rightarrow}{A}}{\partial t}-\sigma\
abla\text{V}. \tag{9}\]  
Because electric scalar potential (V) is realized only when the applied current density (\(J_{s}\)) is established, through the AC steady-state analysis, the differential for time changes to the form is multiplied by vector potential. Equation (9) can be expressed as a relationship between the current density (\(J_{s}\)) applied to an object and the magnetic vector potential (\(\overset{\rightarrow}{A}\)), resulting in:  
\[\left(\frac{1}{\mu}\right)\
abla^{2}\overset{\rightarrow}{A}=-j\omega\sigma \overset{\rightarrow}{A}+\overset{\rightarrow}{J_{s}}. \tag{10}\]  
The results were analyzed using the impedance values of the coils used for the inspection in the actual eddy-current detection test, which measures the total resistance to the current flow at the time of AC power application. The change in impedance of the detected coil is caused by a discontinuity factor, and can be analyzed using the changed impedance value if the specimen to be measured is defective or the aggregate is present.