## 2 Numerical Model  
Eddy Current Testing (ECT) is a nondestructive testing method based on the principle of electromagnetic induction. Usually, a coil with alternating current is employed to generate an alternating magnetic field. When the alternating magnetic field passes through the tested conductor, circular currents, known as eddy currents, are induced within the conductor. The other magnetic field generated by the eddy current affects the original magnetic field, resulting in variations in the magnetic flux in the detection coil. This variation can be measured through the change in impedance in the detection coil. By measuring the change in the impedance in the detection coil, variations in the physical properties of the conductor material can be inferred, including parameters like the carbon content and areal density in CFRP prepre. The equivalent impedance of a coil can generally be represented by the function in Equation (1) [15]:  
\[Z=F(\sigma,\mu,f,x,\lambda) \tag{1}\]  
where \(\sigma\) and \(\mu\) are the electrical conductivity and magnetic permeability of the measured conductor, \(f\) is the frequency of the excitation signal, \(x\) is the distance between the coil and the conductor, and \(\lambda\) is the size factor of the coil. CFRP prepre mainly constitutes of carbon fibers and resin that are unevenly and longitudinally arrayed. When the CFRP prepre is placed within a constant electric field,the majority of electrons shift along the direction of the fibers, while only a small quantity of electrons leap to another fiber via the contact points among the carbon fibers. As a result, it features an extremely complex geometric structure and anisotropy of electrical conductivity. The anisotropy and the complex geometry bring about challenges when it comes to numerical simulation. To circumvent these difficulties, numerous works employ homogenized mathematical expressions to simulate these materials [16]. The electrical conductivity tensor of carbon fiber composite materials is represented in Equation (2) [17]. \[[\sigma]= \begin{bmatrix}\sigma_{\text{L}}cos^{2}(\theta)+\sigma_{\text{T}} sin^{2}(\theta)&\frac{\sigma_{\text{L}}-\sigma_{\text{T}}}{2}sin(2\theta)&0\\ \frac{\sigma_{\text{L}}-\sigma_{\text{T}}}{2}sin(2\theta)&\sigma_{\text{L}} sin^{2}(\theta)+\sigma_{\text{T}}cos^{2}(\theta)&0\\ 0&0&\sigma_{\text{cp}}.\end{bmatrix}  \tag{2}\]  
where \(\sigma_{\text{L}}\) denotes the conductivity along the fiber direction, \(\sigma_{\text{T}}\) represents the conductivity in the transverse direction of the fiber, and \(\sigma_{\text{cp}}\) represents the conductivity in the thickness direction of the plate. \(\theta\) represents the fiber orientation angle. The electromagnetic problem is formulated in terms of the electric vector potential T and the scalar magnetic potential \(\varphi\), which are governed by Equations (3) and (4), involving the conductivity tensor of the CFRP plate, the magnetic permeability of free space \(\mu_{0}\), and the angular frequency \(\omega\). \(\mathbf{H}\) is the magnetic field intensity. Hereinafter, \(V_{s}\) and \(V_{c}\) signify the volumes of the excitation coil and the CFRP plate region, respectively, and \(V_{c}\) is delineated by the boundary. Bold letter are used to represent vectors throughout the article. \[\
abla\times\sigma^{-1}\
abla\times\mathbf{T}+j\omega\mu_{0}(\mathbf{T}- \
abla\varphi)=-j\omega\mu_{0}\mathbf{H} \tag{3}\]  
\[\
abla\cdot(\mathbf{H}+\mathbf{T}-\
abla\varphi)=0 \tag{4}\]  
The current density \(\mathbf{J}\) in \(V_{c}\) is given by Equation (5):  
\[\mathbf{J}=\
abla\times\mathbf{T} \tag{5}\]  
In order to eliminate the normal component of the current density at the \(V_{c}\) boundary, a boundary condition is required, such as that in Equation (6):  
\[\mathbf{n}\times\mathbf{T}=\mathbf{0}/_{\Gamma} \tag{6}\]  
The magnetic field \(\mathbf{H}\) in Equation (3) is calculated using the Biot-Savart Equation (7), where \(\parallel\mathbf{r}-\mathbf{r}^{{}^{\prime}}\parallel\) is the distance between a source of excitation in \(V_{s}\) and \(V_{c}\).