Previously, we developed a method based on FEM and FDTD for the study of an Electromagnetic Acoustic Transducer Array (EMAT). This paper presents a new analytical solution to the eddy current problem for the meander coil used in an EMAT, which is adapted from the classic Deeds and Dodd solution originally intended for circular coils. The analytical solution resulting from this novel adaptation exploits the large radius extrapolation and shows several advantages over the finite element method (FEM), especially in the higher frequency regime. The calculated Lorentz force density from the analytical EM solver is then coupled to the ultrasonic simulations, which exploit the finitedifference timedomain (FDTD) method to describe the propagation of ultrasound waves, in particular for Rayleigh waves. Radiation pattern obtained with Hilbert transform on timedomain waveforms is proposed to characterise the sensor in terms of its beam directivity and field distribution along the steering angle, which can produce performance parameters for an EMAT array, facilitating the optimum design of such sensors.
There are a variety of nondestructive testing (NDT) techniques employed in industries, such as magnetic particle inspection (MPI), electromagnetic methods (EM), eddy current methods, and ultrasonic methods [
Electromagnetic acoustic transducers (EMATs) are becoming increasingly popular due to their noncontact nature [
Considerable works have been reported on EMAT modelling [
Summary of methods used for modelling EMAT.
People  Electromagnetic simulation  Ultrasonic simulation  

FEM  Analytical method  FEM  FDTD  Analytical method  
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The method using finite element method (FEM) and finitedifference timedomain (FDTD) to model EMATs has been reported recently by authors in [
An EMAT sensor consists basically of a coil carrying an alternating current, a permanent magnet providing a large static magnetic field, and the test piece, as shown in Figure
The configuration of a typical EMAT.
The testing sample used is a stainless steel plate with a dimension of 1000 × 1000 × 80 mm^{3}, and the permanent magnet used is NdFeB35, whose size is 80 × 80 × 30 mm^{3}. The meander coil carries an alternating current with the peak of 50 A, the liftoff is 1 mm, the operation frequency is 500 kHz, and the skin depth calculated is 0.679 mm. The Rayleigh wave velocity is 3.033 mm/
For electromagnetic calculation, an analytical solution is adapted from the Deeds and Dodd formula to obtain the magnetic vector potential and the eddy current density. Other analytical solutions are available as well [
Dodd and Deeds proposed analytical solutions to the circular coil over a layered conductor in [
The geometry used in [
The governing equations for induced eddy current calculation are
From (
We build a model to study the analytical solutions to the vector potential problem; the test piece used is stainless steel, and the parameters used are listed in Table
Parameters used for studying the analytical solutions.
Description  Symbol  Value 

The length of the stainless steel 

5 mm 
The height of the stainless steel 

5 mm 
Inside radius of the circular coil 

2.45 mm 
Mean radius of the circular coil 

2.5 mm 
The height of the coil 

1 mm 
Permeability of air 

1.2566 × 10^{−6} H/m 
Liftoff 

1 mm 
Current density 

1 A/m^{2} 
Outside radius of the circular coil 

2.55 mm 
Frequency 

10 kHz 
Conductivity of stainless steel 

1100000 siemens/m 
Permeability of stainless steel 

1.26 × 10^{−6} H/m 
The magnitude distribution of the vector potential
In this work, the coil used in EMAT is a meander coil, so the analytical solutions to a straight wire are needed. Based on the analytical solutions proposed by Dodd and Deeds, we proposed an assumption; that is, when the radius of the circular coil is very large, the bent wire of the circular coil can be approximated to a straight wire, and the distribution of the vector potential would be symmetrical. To verify this assumption, we build a model with the same parameters used in Table
The magnitude distribution of the vector potential
For a straight wire solution, in order to investigate the accuracy of the adapted analytical solutions, the comparison between the analytical solution and the finite element method (FEM) is needed. Maxwell Ansoft is used to construct a model with the same parameters used in Section
The magnitude distribution of the vector potential based on different methods.
Analytical method
Finite element method (FEM)
On the surface of stainless steel (
With an operating frequency of 10 kHz, the distribution of the vector potential along the surface of stainless steel; (a) the magnitude distribution, (b) the real part distribution, and (c) the imaginary distribution.
In this part, the analytical solutions at a high operating frequency are studied. This is because EMAT normally operates at high frequencies and the eddy current is typically limited near the surface. The model used is the same as that in Section
Along the surface of the stainless steel (
With an operating frequency of 1 MHz, the real part distribution of the vector potential along the stainless steel surface. Real part of
The meander coil used in this study has a dimension of 56 × 34.163 × 0.036 mm^{3}, which is very small compared to the size of the stainless steel plate. In order to improve modelling time, only the area (100 × 100 × 2 mm^{3}) where the meander coil mainly has an effect is picked to study the Lorentz force distribution.
As mentioned before, the distribution of the induced eddy current under a straight wire can be obtained by the analytical solutions. For a meander coil, the total induced eddy current is the sum of the induced eddy current caused by each wire segment; the distribution of the induced eddy current on
The distribution of the induced eddy current based on the analytical method. Real part of the induced eddy current.
Along the surface of the stainless steel plate, eddy current distribution is shown in Figure
Fields distribution along the surface of the stainless steel plate; (a) the distribution of the induced eddy current and (b) the distribution of Lorentz force density.
Elastodynamic equations are a set of partial differential equations describing how material deforms and becomes internally stressed as shown in the following [
The finitedifference timedomain (FDTD) method is a numerical method to solve differential equations by discretion of the differential form to the finitedifference form [
In this work, Lorentz force density obtained from the electromagnetic model is imported to ultrasonic model to generate ultrasound waves. As shown in Figure
Transformation from electromagnetic model to ultrasonic model.
The propagation of ultrasound waves is shown in Figure
Wave propagation at 23
Wave propagation at 45
The receiving signals from R1 and R2 are shown in Figure
Receiving signals from receivers R1 and R2.
The receiving coil used is the same as the transmitting coil; the induced voltage in the receiving coil can be calculated from the receiving velocity fields [
Most of the previous works calculate the radiation pattern based on analytical equations [
The amplitude and envelope of receiving signals from R1.
The radiation pattern of the EMATRayleigh waves is shown in Figure
The radiation pattern of EMATRayleigh waves for studying beam features.
The radiation pattern of EMATRayleigh waves
Radiation pattern for beam features
There are two beam features to be analysed, beam directivity and field distribution along the steering angle. Beam directivity is, at a radial length from the centre of the array, the velocity or pressure distribution, as shown in the green circle arc in Figure
The beam directivity and field distribution along the steering angle 0° of EMATRayleigh waves.
Another beam feature, field distribution along the steering angle, is studied as well. The radial distance
A method combining the analytical method for EM simulation and the finitedifference timedomain (FDTD) method for UT simulation to model an EMAT system is proposed. For electromagnetic simulation, analytical solutions to a meander coil are proposed and verified with FEM. By comparing with FEM, the analytical method proposed provides several advantages, especially at high frequencies. The calculated Lorentz force density is used as the excitation source in the ultrasonic simulation, which exploits the finitedifference timedomain (FDTD) method to describe ultrasound wave propagation. The radiation pattern shows that the maximum energy of surface waves is concentrated at the steering angle 0° and at the position where the sensor is placed.
The authors declare that there is no conflict of interests regarding the publication of this paper.