# A Pseudo-3D Model for Electromagnetic Acoustic Transducers (EMATs)
## 1 Introduction  
A wide group of non-destructive testing (NDT) techniques are commonly used in biomedical industries, such as ultrasonic techniques, electromagnetic techniques, and laser testing [1; 2; 3; 4]. Due to the non-contact nature, more and more attention has been paid to the NDT technique with electromagnetic acoustic transducers (EMATs), and EMATs have gradually been used in industrial applications, such as thickness gauging and defect detection [5; 6; 7; 8]. A classic EMAT sensor is made of a meander-line-coil and a permanent magnet (Figure 1). There are two major coupling principles for EMATs: magnetostriction is for ferromagnetic metallic materials, and the Lorentz force mechanism is for conductive and ferromagnetic materials [9]. This work focussed on only the Lorentz force mechanism performing on an aluminium plate. The Lorentz force mechanism is: the meander-line-coil placed above the sample generates eddy currents \(\mathbf{J}\) within the sample. A permanent magnet placed above the coil generates a static magnetic field \(\mathbf{B}\) to the sample. The interaction between \(\mathbf{J}\) and \(\mathbf{B}\) produces Lorentz force density \(\mathbf{F}\), as shown in Equation (1):  
\[\mathbf{F}=\mathbf{J}\times\mathbf{B} \tag{1}\]  
Substantial works have been reported on EMATs modelling, which comprises an electromagnetic (EM) model and an ultrasonic (US) model [10; 11; 12]. The EM model was accomplished by the finite element method (FEM) and the analytical method, while the US model was accomplished with FEM, the finite-difference time-domain method (FDTD), and the analytical solutions. Some of the previouswork modelled EMATs by combining FEM and analytical solutions, i.e., FEM for EM modelling, and analytical solutions for US modelling [10; 11; 12]. On the other hand, some of the previous work utilised FEM for both EM and US modelling, i.e., COMSOL (a commercial EM simulation package) for EM modelling, and Abaqus for US modelling [13; 14]. Authors have proposed several methods to model EMATs, including a method combining FEM and FDTD, a method combining analytical solutions and FDTD, and a wholly analytical method [15; 16; 17; 18]. Most of EMATs' simulations were two-dimensional (2D), and can only focus on the specific plane. This article is attempting to build a pseudo-three-dimensional (3D) model in order to further study EMATs by combining a surface plane 2D model (the \(x\)-\(y\) cross-section shown in Figure 1) and a vertical plane 2D model (the \(y\)-\(z\) cross-section shown in Figure 1) together. More specifically, the Lorentz force density obtained from the vertical plane of the sample is imported to the surface plane of the sample as the driving source to generate Rayleigh waves. Previously, only the beam directivity at the surface of the sample (\(z\) = 0 in Figure 1) was investigated [18; 19]. However, Rayleigh waves not only distribute along the surface (\(z\) = 0), they also distribute within a depth of the Rayleigh waves' wavelength. Some industrial defects are within the sample instead of on the surface of the sample; in order to use Rayleigh waves to detect such defects, the beam directivity of the Rayleigh waves at various depths is worth investigating. In this article, the equations to study the beam directivity are more complete compared to the approximate equations presented in [18] by [PERSON] et al., and the beam directivity at various depths are investigated. This study lays a solid industrial foundation for near-surface defects detection using Rayleigh waves, and can be a starting point to build an advanced 3D EMAT model in the future. Except for near-surface detection, the proposed strategy can be used to perform body detection, i.e., to model bulk waves, including longitudinal waves and shear waves. In addition, the proposed 3D EMAT model can be used to characterise other EMAT structures to generate surface waves, such as unidirectional Rayleigh waves EMATs and multiple directional Rayleigh waves EMATs.