# Enhanced Nondestructive Testing Using Pulsed Eddy Current and Electromagnetic Ultrasonic Techniques
## 2 Pulsed Eddy Current and Electromagnetic Ultrasonic Testing
### Principles of Electromagnetic Ultrasonic Testing  
An electromagnetic ultrasonic transducer, a key component of electromagnetic ultrasonic testing technology, represents a nondestructive testing method that uses electromagnetic effects to excite and capture ultrasonic waves. The transducer comprises three main components: a coil that receives high-frequency excitation, a biased magnetic field emanating from the test piece, and a magnet. Depending on the coil structures and magnetic field orientations, the electromagnetic ultrasonic transducer can produce various types of ultrasonic waves, making it versatile for different applications. Transducers that generate ultrasonic body waves are predominantly used for measuring material thickness and are particularly effective in detecting corrosion, thinning defects, and internal flaws within materials.  
Ultrasonic body waves are categorized into shear waves and longitudinal waves. Typically, the velocity of ultrasonic shear waves is half that of longitudinal waves in solids, resulting in a higher time resolution in shear wave detection. This study, therefore, employs a combined approach using ultrasonic S-wave detection and pulsed eddy current detection. The principle of ultrasonic S-wave detection is illustrated in Figure 2.  
Figure 1: Schematic of pulsed eddy current detection.  
At specific time \(t_{1}\), the coil is activated, inducing eddy currents on the conductor's surface. These currents cause particles on the surface to oscillate horizontally owing to the vertical external magnetic field, which, in turn, propagates ultrasonic shear waves through the conductor. The particle vibrations can be expressed by the following equation:  
\[(\lambda_{a}+2G_{a})\
abla(\
abla\cdot u)-G_{a}\
abla\times\
abla\times u+f_{L}= \rho\frac{\partial^{2}u}{\partial t^{2}}. \tag{3}\]  
In this equation \(\lambda_{a}\), \(G_{a}\), and \(\rho\) represent the Gabriel Lame constant, shear modulus, and density of the conductor, respectively; u represents the particle displacement; and \(f_{L}\) represents the density of the Lorentz force on the surface of the material.  
As the wave encounters a defect or reaches the bottom of the conductor, it reflects back to the surface, inducing a voltage in the coil:  
\[\left\{\begin{array}{l}\frac{1}{\mu}\
abla^{2}A-\sigma\frac{ \partial A}{\partial t}+\frac{\sigma}{S_{c}}\frac{\partial}{\partial t} \underset{S_{c}}{\iint}\,Ad_{s}=-J_{L}\\ V_{out}=\int-\frac{\partial A}{\partial t}\cdot dl\end{array}\right., \tag{4}\]  
where \(J_{L}\) denotes the induced eddy current density caused by the vibrating particles on the material surface under the influence of the external magnetic field, \(S_{c}\) is the area of the receiving coil, and \(V_{out}\) represents the induced voltage on the receiving coil.  
The process of signal reception is considered the inverse of transmission, with the current time denoted by \(t_{2}\). The depth of the defect (\(h\)) is then determined based on the propagation velocity (\(v_{s}\)) and propagation time (\(\Delta t=t_{2}-t_{1}\)) of the shear wave in the conductor:  
\[h=\frac{1}{2}v_{s}\Delta t=\frac{1}{2}v_{s}(t_{2}-t_{1}). \tag{5}\]