## 4 Theoretical Study
### Electromagnetic Simulations  
Both the coil design and the magnet configuration affect the distribution of Lorentz force in space, its direction, and its amplitude. As a result, the wave mode the EMAT is to generate and/or receive gets affected by the configuration of its two main components. An EMAT configuration that can be used for the SH\({}_{0}\) is the PPM EMAT with racetrack coil [26; 27; 28; 29; 30]. In a PPM EMAT, the distance between two adjacent magnets whose magnetic field has the same direction (pitch) is equal to the wavelength; the arrangement of the magnets is illustrated in Figure 4. Racetrack coil is also used in this EMAT configuration. In this coil design there are no gaps within its turns, resulting in its broadband response in frequency. Hence, only the magnets' arrangement and the frequency of the AC current can affect the frequency response of EMAT. Finite Element Analysis (FEA) was used for the theoretical validation of this EMAT design regarding its guided wave purity characteristics at room temperature and at 180 \({}^{\circ}\)C. A coupled electromagnetic and mechanical analysis was carried out in COMSOL. A 3D model was created, in which a 12-magnet PPM EMAT was evaluated regarding eddy current, magnetic flux, and Lorentz force distribution. In this model two arrays of six Nd-Fe-B magnets each was simulated; each magnet has a 15 mm width, 5 mm depth, 5 mm height, and magnetization of 750 kA/m, and the direction of their magnetic flux is on the \(z\) axis. Their arrangement in space is the same as Figure 4 and, thus, the distance between one another is 1 mm. Due to the high requirements in time and computational power these complicated models have, the copper coil has been simplified and designed as two rectangular blocks of 35 mm width, 15 mm depth, and 0.4 mm height. The AC current driven to the coil is a two-cycle, Hanning windowed sinusoidal wave of 20 A amplitude and 256 kHz central frequency with \begin{table}
\begin{tabular}{c|c|c|c|c|c|c|c|c} \hline \hline \multicolumn{6}{c|}{**Dispersion Curves—SH/S/A/A/A/A Frequency \& Wave Velocity**} \\ \hline
**Value** & \multicolumn{6}{c|}{**Frequency (kHz)**} & \multicolumn{6}{c}{**Velocity (m/s)**} \\ \hline Temperature (\({}^{\circ}\)C) & 22 & 60 & 100 & 180 & 22 & 60 & 100 & 180 \\ \hline SH\({}_{0}\) & 256 & 255 & 254.2 & 250.6 & 3080 & 3067 & 3051 & 3008 \\ \hline S\({}_{0}\) & 420 & 419 & 415.5 & 405 & 5044 & 5022 & 5003 & 4944 \\ \hline A\({}_{0}\) & 143 & 140 & 139 & 136 & 1736 & 1720 & 1711 & 1685 \\ \hline A\({}_{1}\) & 676 & 673 & 668 & 657 & 8243 & 8217 & 8203 & 8129 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dispersion Curves, wave velocity, and frequency. Figure 4: **(a)** Magnet arrangement in a PPM EMAT; **(b)** racetrack coil. direction on the \(x\) axis. The orientation of the excitation current flowing inside one rectangular coil is opposite to the orientation of the current inside the other coil. The EMAT has a lift-off of 0.6 mm from the specimen, which is a 316 L stainless steel plate of 3 mm thickness, 750 mm width, and 750 mm depth and of the same material properties as the plate simulated in the previous section at room temperature and at 180 \({}^{\circ}\)C. The material, magnetic, and electrical properties of the EMAT did not change with temperature. Figure 5a,b show the excitation/eddy current at 4 \(\upmu\)s and the static magnetic flux distribution, respectively. Both the electric and the magnetic field are uniformly distributed. The orientation of the eddy current alters between the two sides of the coil and it is on the _x-y_ plane. The orientation of the magnetic field alters as it is depicted in Figure 5b; its orientation is mainly on the \(z\) axis and its maximum strength is observed at the center of each magnet separately, as is expected. Thus, the Lorentz force generated should mainly result in an in-plane displacement. A probe was also placed 30 cm away from the EMAT for obtaining the in-plane (_x-y_ plane) and out-of-plane (_x-z_ plane) displacement.