# A Pseudo-3D Model for Electromagnetic Acoustic Transducers (EMATs)

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## Abstract

**:**

## 1. Introduction

**J**within the sample. A permanent magnet placed above the coil generates a static magnetic field

**B**to the sample. The interaction between

**J**and

**B**produces Lorentz force density

**F,**as shown in Equation (1):

## 2. Vertical Plane Modelling

#### 2.1. EMAT-EM Model

#### 2.1.1. Adapted Analytical Solutions to the Vector Potential for a Straight Wire

**A**is the vector potential generated by

**I**, ω and

**I**are the angular frequency and the density of the applied alternating current (AC), respectively, ε, µ and σ are the permittivity, permeability and conductivity of the test piece respectively, and

**E**and

**J**are the induced electric field and eddy current density, respectively [20].

**A**at z = 0 (surface of the sample) is not symmetrical with the radius due to the bent wire.

**A**should be symmetrical. To validate such a hypothesis, a model is built with a large-radius circular coil above the aluminium plate. The aluminium sample used has a dimension of 80 mm × 30 mm, and the inner radius and the outer radius of the circular coil are set to 5.0395 m and 5.0405 m, respectively. At 1 kHz, the current density applied to the circular coil is 1 A/m

^{2}, and the lift-off of the coil is 1 mm. The permeability and the conductivity of the aluminium plate are 1.257 × 10

^{−6}H/m and 3.8 × 10

^{7}Siemens/m, respectively.

**A**based on the adapted solution is shown in Figure 2. The red marker denotes the maximum magnitude of the vector potential. The distribution of the magnitude of

**A**is symmetrical with the radius of 5.04 m, where the coil is located. The result verifies the hypothesis that, when the radius of the circular coil is very large, a bent wire serves as a straight wire.

#### 2.1.2. Comparison between the Adapted Solution and FEM

**A**at the sample’s surface (z = 0) is presented in Figure 3. The analytical solution and FEM present a good agreement at an operational frequency of 1 kHz. However, at a working frequency of 1 MHz, the distribution of

**A**from the FEM is not smooth compared to that from the analytical solution; the reason is that the FEM is affected by the elements density and numerical approximation is unavoidable, etc. Therefore, the adapted analytical method presents a more accurate result compared to FEM, especially for a high working frequency.

#### 2.1.3. EMAT-Lorentz Force Calculation

**A**(vector potential) generated by a meander-line-coil is the addition of

**A**generated by every single straight wire. The zone where the meander-line-coil mainly operates on is selected to model EM simulation to increase the modelling effectiveness. The distribution of

**A**and

**F**at z = 0 are shown in Figure 4 and Figure 5. The generated periodic fields have different directions for any neighbouring wires, since their applied ACs are opposite, and therefore, the periodic distribution of

**A**and

**F**has six positive values and six negative values. The outermost

**A**and outermost

**F**are largest, because

**A**is under the outermost wires, and thus is only determined by the fields on one side.

#### 2.2. EMAT-US Simulation

#### 2.2.1. Elastodynamic Equations

#### 2.2.2. Combination of EMAT-EM and EMAT-US Models

**F**is obtained from the EMAT-EM calculation. In this section,

**F**, which is used as the force source, is imported to the EMAT-US model to produce ultrasound (Figure 6). Since

**F**is calculated in the frequency domain and FDTD is a time-domain solver, the excitation signal for the EMAT-US model is a time sequence signal with the peak equalling the peak values of

**F**. The excitation signal used is a Gaussian-modulated sinusoidal with a fractional width of 0.18. A crack and a receiver R are located within the sample, as shown in Figure 6. Regarding the FDTD setup in the ultrasonic model, there are two main parameters: the spatial step, and the time step. The spatial step used is 0.2 mm, which approximately equals to 1/30th of the wavelength. The time step is 0.0222 µs, which is calculated based on the Courant–Friedrichs–Lewy (CFL) condition. Free surface conditions are utilised on the surface of the sample. Perfectly-matched layers (PML) with a thickness of 16 mm are utilised to absorb ultrasound.

#### 2.2.3. Wave Propagations

#### 2.3. EMAT-Reception Simulation

#### 2.4. Experimental Validations

## 3. Horizontal Surface Plane Modelling—Directivity Analysis of Rayleigh Waves

#### 3.1. The Analytical Solution to the Displacement of Rayleigh Waves

#### 3.2. Linking EMAT-EM and EMAT-US Models

**F**values, acting as the excitation source, are imported to each surface layer at various depths. The Rayleigh waves’ distribution is the addition of the Rayleigh waves generated by each point source. Table 1 illustrates the parameters used for the EMAT-US model.

#### 3.3. Analysis of the Beam Directivity of Rayleigh Waves

_{1}= −70°, and θ

_{2}= 70°).

#### 3.4. Experimental Validations

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**The configuration of a typical meander-line-coil electromagnetic acoustic transducer (EMAT). Reproduced with permission from [15], IEEE, 2016.

**Figure 2.**The magnitude distribution of A under a large-radius circular coil. Reproduced with permission from [15], IEEE, 2016.

**Figure 3.**The distribution of A from the adapted analytical solution and the finite element method (FEM). The left curves are the results at 1 kHz, while the right curves are the results at 1 MHz. The red curve is the result from the FEM, and the blue curve is the result from the analytical solutions. Reproduced with permission from [15], IEEE, 2016.

**Figure 4.**For a meander-line-coil, the distribution of the vector potential

**A**at z = 0. Reproduced with permission from [15], IEEE, 2016.

**Figure 5.**For a meander-line-coil, the distribution of the Lorentz force density

**F**at z = 0. Reproduced with permission from [15], IEEE, 2016.

**Figure 6.**On the vertical plane of the material, the combination of the EMAT-electromagnetic (EM) and EMAT-ultrasonic (US) models. Reproduced with permission from [15], IEEE, 2016.

**Figure 7.**Wave propagations. (

**a**,

**b**) denote the velocity fields at 27 µs and 83 µs, respectively. Reproduced with permission from [15], IEEE, 2016.

**Figure 8.**The received signal from simulations. Reproduced with permission from [15], IEEE, 2016.

**Figure 9.**The received signal from experiments. The blue curve denotes the induced voltage in the received coil, and the red curve denotes the envelope of the received signal. Reproduced with permission from [15], IEEE, 2016.

**Figure 10.**The envelope of the received signals. The blue curve and the red curve are the envelope of the received signal from simulations and experiments, respectively.

**Figure 11.**The model used to simulate Rayleigh waves. Reproduced with permission from [18], Elsevier, 2017.

**Figure 12.**On the surface plane of the material, the transformation between the EM and US models. Reproduced with permission from [18], Elsevier, 2017.

**Figure 13.**The radiation pattern of Rayleigh waves generated by the meander-line-coil EMAT. Reproduced with permission from [18], Elsevier, 2017.

**Figure 14.**The simulated beam directivity of Rayleigh waves at z = 0. Reproduced with permission from [15], IEEE, 2016.

**Figure 17.**The comparison between the simulated beam directivity and the experimental beam directivity at z = 0. The red curve is the beam directivity from simulations, while the blue curve is the beam directivity from experiments.

Description | Symbol | Value |
---|---|---|

Length of the aluminium plate | Y | 600 mm |

Width of the aluminium plate | X | 600 mm |

Field spatial step | ∆x_{f} | 1 mm |

Length of the meander-line-coil | L | 50 mm |

Source spatial step for each wire | ∆x_{s} | 0.2 mm |

Density of the aluminium plate | ρ | 2700 kg/m^{3} |

Frequency | f | 483 kHz |

Longitudinal waves’ velocity | C_{l} | 6.375 mm/$\mathsf{\mu}\mathrm{s}$ |

Shear waves’ velocity | C_{s} | 3.14 mm/$\mathsf{\mu}\mathrm{s}$ |

Rayleigh waves’ velocity | C_{r} | 2.93 mm/$\mathsf{\mu}\mathrm{s}$ |

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**MDPI and ACS Style**

Yin, W.; Xie, Y.; Qu, Z.; Liu, Z.
A Pseudo-3D Model for Electromagnetic Acoustic Transducers (EMATs). *Appl. Sci.* **2018**, *8*, 450.
https://doi.org/10.3390/app8030450

**AMA Style**

Yin W, Xie Y, Qu Z, Liu Z.
A Pseudo-3D Model for Electromagnetic Acoustic Transducers (EMATs). *Applied Sciences*. 2018; 8(3):450.
https://doi.org/10.3390/app8030450

**Chicago/Turabian Style**

Yin, Wuliang, Yuedong Xie, Zhigang Qu, and Zenghua Liu.
2018. "A Pseudo-3D Model for Electromagnetic Acoustic Transducers (EMATs)" *Applied Sciences* 8, no. 3: 450.
https://doi.org/10.3390/app8030450