## 2 Principle and Methods  
The material characteristics technology with eddy current method is based on the constant phase angle pulse eddy current (CPA-PEC) sensor. The metal material properties sample is formed by extracting the frequency variation and attenuation characteristics of the eddy current field using a discrete wavelet transform model. The CPA-PEC sensor is an eddy current sensor with a dual coil probe structure, as shown in Figure 1, in which the excitation coil is fed with a periodic ramp signal with a constant change in current to generate the magnetic field. The receiving coil is placed outside the excitation coil to measure the change in the magnetic field. According to the law of electromagnetic induction, it is known that the voltage across the receiving coil is proportional to the rate of change of the magnetic flux at the coil location. The magnetic flux at the receiving coil is related to the material, thickness, lift-off and shape of the detected metal. Different frequencies of the magnetic field in the metals to be measured attenuate the energy differently, so that the corresponding time domain signal in the receiving coil changes. It can then be extracted from the detection target material, thickness, lift-off, shape and other information through data processing. The working principle is described as follows. The model is simplified for better analysis: the coils are considered as a clingy excitation-receiving system without volume. The excitation source used in this paper is a ramp signal as shown in Figure 1, i.e., \(I_{SE}=k_{1}t\), where \(k_{1}\) is constant. Then, the magnetic field at its axial position x from the center is  
\[B=\frac{N_{1}\mu_{0}I_{SE}R^{2}}{2{(R^{2}+x^{2})}^{3/2}} \tag{1}\]  
where \(N_{1}\) is the turns of the excitation coil, \(\mu_{0}\) is the vacuum magnetic permeability and \(R\) is the coil radius; \(I_{SE}\) is the current amplitude of the excitation source. Then, the voltage across the induction coil \(U_{s}\) should be  
\[U_{s}=\frac{N_{2}d\varnothing}{dt}=\frac{N_{1}N_{2}S\mu_{0}R^{2}k_{2}}{2{(R^{2 }+x^{2})}^{3/2}} \tag{2}\]  
Figure 1: The structure of CPA-PEC sensor. where \(\Phi\) is the magnetic flux flows through the coil, \(S\) is the induction area of the coil and \(N_{2}\) is the number of turns of the receiving coil. \(k_{2}\) is a scale factor of the central magnetic field to the detected magnetic field. From Equation (2), it can be seen that the voltage in the induction coil should be constant when the excitation field is a ramp signal. That is, in free space, the voltage across the induction coil is a constant value. Assuming an infinite plane current magnetic field source with a magnetic field \(H_{0z}\), the solution for the magnetic field in the conductor located at \(x\) is  
\[H_{z}=H_{0z}e^{-\frac{1+\psi}{\sqrt{2}}\sqrt{\alpha\mu\rho}x} \tag{3}\]  
where \(H_{z}\) is the z-directional component of the magnetic field in the metal, \(x\) is the location of \(H_{z}\) in the conductor, \(\omega\) is the angular frequency, \(\mu\) is the magnetic permeability and \(\sigma\) is the electrical conductivity. It is clear that the current inside the conductor is related to the magnetic field frequency \(\omega\) and the conductor's electromagnetic properties \(\mu\), \(\sigma\). The intensity of eddy currents in the conductor increases with frequency within a certain range, but the eddy currents become more and more concentrated in the center of the magnetic field with increasing frequency. Since the magnitude of the magnetic field is proportional to the product of the induced current and its loop area, the magnetic field decreases with increasing frequency within a certain range. It means that there is an optimal magnetic field frequency in detection. The optimal frequency point is dependent of the conductor properties \(\mu\), \(\sigma\) in Equation (1). The wavelet decomposition of the ramp signal shows that the ramp signal corresponds to different spectra at different times in the time domain.