The wave structure refers to the distribution of the physical quantities such as displacement, stress, and strain along the thickness of the plate. The wave structure characteristics of the Lamb wave show the vibration of the particles along the thickness direction and the Lamb wave energy distribution.^[15] Combined with the characteristic equation of Lamb wave and wave structure theory, the normalized displacement wave structure of aluminum Lamb wave can be obtained.^[15] Figure 1 shows the normalized displacement wave structures of Lamb wave A0 and S0 mode at 400 kHz in a 3-mm-thick aluminum plate with a longitudinal speed of 6350 m/s and a transverse speed of 3100 m/s. It can be seen from Fig. 1(a) that in the plate thickness range, u_x is symmetrical with respect to the origin, and u_z is symmetrical with respect to the straight line where the displacement of plate thickness is taken as 0. The u_x and u_z represent the in-plane displacement and the out-of-plane displacement, respectively, indicating the displacement of the Lamb wave in the propagation direction of the aluminum plate and the displacement of the Lamb wave in the thickness direction of the aluminum plate. The magnitude of u_z is greater than u_x. The u_x reaches a maximum on the upper and lower surfaces of the plate, and it is zero at the center of the plate. It can be seen from Fig. 1(b) that in the plate thickness range, u_z is symmetrical with respect to the origin, and u_x is symmetrical with respect to the straight line where the displacement of plate thickness is taken as 0. The magnitude of u_x is greater than u_z. The u_z reaches the maximum on the upper and lower surfaces of the plate, and it is zero at the center of the plate.

Fig. 1.

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	Fig. 1. (color online) Normalized displacement wave structures of Lamb wave at 400 kHz in 3 mm-thick aluminum plate: (a) A0 mode, and (b) S0 mode.

The amplitude of in-plane displacement of the S0 mode is large, and the out-of-plane displacement is almost zero. The A0 mode is dominated by out-of-plane displacement, and both the in-plane displacement and out-of-plane displacement amplitude of the A0 mode are large. Therefore, the perpendicular magnetic field can more easily generate the S0 mode, and the horizontal magnetic field can more easily generate the A0 mode. This is of great significance for selecting the mode and improving the efficiency of electromagnetic acoustic transduction.

When the EMAT technique is operated on a non-ferromagnetic material, only the Lorentz force mechanism needs to be considered. The two-dimensional model of the meander-coil EMAT that generates Lamb waves in an aluminum plate is shown in Fig. 2, where the thickness of the aluminum plate is 2d, the oz axis indicates the Lamb wave propagation direction, and the oy axis indicates the direction of the plate thickness. The permanent magnet is placed above the meander coil and the static bias magnetic field B_s is applied to the meander coil. Figure 2(a) shows that a perpendicular static bias magnetic field generated by the permanent magnet is applied to the meander coil, and the Lorentz force generated by the perpendicular static magnetic field is along the horizontal direction. Figure 2(b) shows the horizontal static bias magnetic field generated by the permanent magnets is applied to the meander coil, and the Lorentz force generated by the horizontal static magnetic field is along the perpendicular direction. The parameter a represents the width of the meander coil single wire, D denotes the spatial period of the meander coil, and L refers to the total length of the meander coil. The parameter g represents the lift-off distance and is assumed to remain constant throughout the analyses.

Fig. 2.

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	Fig. 2. Two-dimensional model of EMAT for generating Lamb waves: (a) perpendicular static bias magnetic field, and (b) horizontal static bias magnetic field.

An alternating current i(t) = Iexp(jωt) is applied to the meander coil, where the angular frequency ω = 2πf, f is the driving frequency. It is assumed that the Lorentz force P_zy becomes zero outside the EMAT coverage (z ∈ (−L/2,L/2)). The spatial Fourier transform of P_zy is given by (omitting the factor exp(jωt)):^[3,4,16]

In the numerical analyses, the thickness of the aluminum plate is 3 mm. When the driving frequency f is 510 kHz, the phase velocity c_p of A0 mode is 2554 m/s and the phase velocity c_p of S0 mode is 5130 m/s. According to the phase velocity and the driving frequency, we can obtain the spacing between adjacent wires generating a single A0 mode to be c_p/2f = 2554/(2 × 510 kHz) = 2.5 mm and the spacing between adjacent wires generating single S0 mode to be 5 mm. The corresponding D values are 5 mm and 10 mm. Because the ultrasonic wave amplitude reaches its maximum value when the magnet length is three times the meander-coil length,^[14] we choose the meander-coil length to be 15.2 mm (the length of magnet is 40 mm). We separately calculated and analyzed two meander-coils with a = 0.2 mm, L = 15.2 mm, and g = 0.5 mm, but D = 5 mm and 10 mm. When D = 5 mm, we call its corresponding coil No. 1, and when D = 10 mm, we call its corresponding coil No. 2.

When the geometrical parameters of the meander coils are given, the curve of the spatial Fourier transform, denoted by , of the Lorentz force P_zy = P_zy(z) exerted on the aluminum plate (y = +d) is calculated by Eq. (1) and shown on the left side of Figs. 3(a) and 3(b), and n is selected to be 7.^[4] The dispersion curves (wave-number k versus f) of Lamb wave in a 3-mm thick aluminum plate are simultaneously shown on the right side of each of Figs. 3(a) and 3(b). Both the curve and the Lamb wave dispersion curve have the same wave-number axis. The displacement field of the n-th Lamb wave mode is proportional to the magnitude of the spatial transform of P_zy(z) under k = k_n. According to Fig. 3, the magnitudes of the n-th Lamb wave modes corresponding to different given frequencies can be obtained.

Fig. 3.

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	Fig. 3. (color online) Curves of the spatial Fourier transform of the Lorentz force (left) and Lamb wave dispersion curves (right) in each panel: (a) EMAT with coil No. 1, and (b) EMAT with coil No. 2.

It can be seen from Fig. 3(a) that when the driving frequency of the meander coil No. 1 is f = 510 kHz (given by the red dashed line), the Lamb wave A0 and S0 modes may be generated. The corresponding magnitudes of that contribute to the generation of the A0 and S0 modes, are determined by points P₁ and P₂ in Fig. 3(a). The meander coil No. 1 is designed based on the phase velocity of the A0 mode at the driving frequency f = 510 kHz, and the amplitude of the A0 mode is close to the main lobe peak at this moment. However, the amplitude of S0 mode is also large at the same time. When the driving frequency is adjusted to f = 400 kHz (given by the green dashed line in Fig. 3(a)), the A0 and S0 mode may be generated. The corresponding magnitudes of that contribute to the generation of the A0 and S0 modes are determined by points P₃ and P₄ in Fig. 3(a). Clearly, the magnitude of for the generation of the A0 mode (determined by point P₃) is far larger than that of the S0 mode (determined by point P₄). Therefore, the A0 mode will play a dominant role and the S0 mode will be restrained at the frequency f = 400 kHz. When the driving frequency of the meander-coil No. 2 is f = 510 kHz (given by the red dashed line in Fig. 3(b)), the Lamb wave S0 and A0 mode will be also simultaneously generated. The corresponding magnitudes of that contribute to the generation of the A0 and S0 modes, are determined by points P₅ and P₆ in Fig. 3(b). The meander coil No. 2 is designed based on the phase velocity of the S0 mode at the driving frequency f = 510 kHz and the amplitude of S0 mode is close to the main lobe peak at this time. However, the amplitude of the A0 mode is also large at the same time. This does not exert a very good suppression effect on the A0 mode. It can be seen from Fig. 3(b) that when the driving frequency is adjusted to f = 400 kHz, the magnitude of for the generation of the S0 mode (determined by point P₇) is much larger than that of the A0 mode (determined by point P₈), and it can be considered that there will be only the S0 mode, and the A0 mode will be suppressed at this time.

In this subsection, the magnetic flux density profiles of the conventional and the new magnetic configuration will be obtained by simulation. The three-dimensional (3D) finite element model (FEM) is established with Maxwell. The conventional magnetic configuration is generally used as EMATs in combination with meander-coils as shown in Fig. 4(a). The improved EMAT is designed as shown in Fig. 4(b), and the magnetic configuration consists of two identical square magnets side by side with opposite polarity. Both EMATs use the meander-coils with the same length L and width b, and the same magnetic material, NdFeb (grade: N35).

Fig. 4.

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	Fig. 4. Comparison of the magnetic configuration of EMATs between (a) conventional EMAT, and (b) improved EMAT with the new magnetic configuration.

The space volume (w × w × h) of the magnet in the conventional EMAT is the same as the space volume of the magnet in the improved EMAT. The dimension of the magnet in the conventional EMAT is 40 mm × 40 mm × 20 mm, and the dimension of each magnet in the improved EMAT is 40 mm × 20 mm × 20 mm. The residual magnetic flux intensity in magnets is set to be 1.18 T for both cases. Figures 5(a) and 5(b) show the magnetic flux density profile on a 2-mm line just below the center of the conventional and new magnetic configurations. It can be seen from Fig. 5(a) that the horizontal component of the magnetic flux density B_sx is equal to zero at the center and reaches two peaks near the two edges of the magnet. The perpendicular component of the magnetic flux density B_sz has a flat peak under the magnet surface. Therefore, the total magnetic flux density B_s is slightly larger near the edges of the magnet. It can be seen from Fig. 5(b) that the horizontal component of the magnetic flux density B_sx has a maximum at the center of the magnets, and two local peaks at the two edges of the magnets. The perpendicular component of the magnetic flux density B_sz is zero at the center and has two flat peaks under each magnet surface. Therefore, the total magnetic flux density B_s has a large peak at the center and two local peaks at the two edges of the magnets. According to the magnet simulation results, the conventional EMAT magnetic field direction is mainly perpendicular to the meander coil. The magnetic flux density of the magnet in the improved EMAT has both horizontal and perpendicular magnetic flux on the meander coil, and the horizontal magnetic field is stronger. It can also be observed that in the central area where the meander coil is located, the magnetic flux density of the improved EMAT magnet has a much larger peak in Fig. 5(b) than the maximum in Fig. 5(a).

Fig. 5.

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	Fig. 5. (color online) Modelled magnetic flux density curves for (a) conventional and (b) new EMAT magnetic configuration.

According to the wave structure analysis results (shown in Fig. 1), the A0 and S0 modes may be generated by the EMAT transmitter at the driving frequency f = 400 kHz. When the conventional magnet and the coil No. 1 are combined into an EMAT, the magnitude of the generated A0 mode signal will be larger than that of the S0 mode, which is shown in Fig. 3(a). When the conventional magnet and coil No. 2 are combined into an EMAT, the magnitude of the generated S0 mode signal will be larger than that of the A0 mode in Fig. 3(b). Figures 7(a) and 7(b) show the time-domain ultrasonic pulse signals detected by the conventional magnetic configuration with coil No. 1 and coil No. 2 when the spatial separation between the centers of the two transducers (transmitter and receiver) is set to be 300 mm.

Fig. 7.

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	Fig. 7. (color online) Conventional EMATs with (a) coil No. 1, and (b) coil No. 2 as transmitter and receiver at driving frequency f = 400 kHz.

There is one ultrasonic pulse signal in Fig. 7(a) or Fig. 7(b), and the corresponding arriving times are around 110 μs and 80 μs. When the driving frequency f = 400 kHz, the theoretical group velocities of the A0 and S0 modes are 3142 m/s and 4923 m/s, respectively. Thus, the theoretical arriving times of the A0 and S0 mode signal should be 96 μs and 61 μs. Because the burst wave signal has a time delay about 12 μs, the signal in Fig. 7(a) should be the A0 mode signal and the signal in Fig. 7(b) should be the S0 mode signal.

Figure 8 shows the time-domain ultrasonic pulse signals detected by the new magnetic configuration with coil No. 1 and No. 2. There are two ultrasonic pulse signals in Fig. 8(a) and one ultrasonic pulse signal in Fig. 8(b). The corresponding arriving times are around 80 μs and 110 μs in Fig. 8(a) and the corresponding arriving time is around 80 μs. When the driving frequency f = 400 kHz, the theoretical arriving times of the A0 and S0 mode signal should be 96 μs and 61 μs. Because the burst wave signal has a time delay of about 12 μs, the signal in Fig. 8(a) should contain the S0 mode and A0 mode, and the signal in Fig. 8(b) should be the S0 mode signal.

Fig. 8.

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	Fig. 8. (color online) Improved EMATs with (a) coil No. 1, and (b) coil No. 2 as transmitter and receiver at driving frequency f = 400 kHz.

The Lamb wave S-series mode (S0, S1, S2, . . .) is dominated by in-plane displacement, and the A-series mode (A0, A1, A2, . . .) is dominated by out-of-plane displacement. Therefore, the perpendicular magnetic field can more easily generate the S-series mode, and the horizontal magnetic field can more easily generate the A-series mode. Since coil No. 1 is designed based on the phase velocity of the A0 mode, the horizontal magnetic field plays a dominant role regardless of the magnetic configuration. Similarly, since coil No. 2 is designed based on the phase velocity of the S0 mode, the perpendicular magnetic field plays a major role. Through the analysis of the above experimental results, we can see that the experimental conditions in Figs. 7(b) and 8(b) are the same, except that the magnetic configuration is different. Both magnetic configurations have the same space volume, and the meander coil is placed in the center of the magnetic configuration and the coil length is 1/3 of the magnet length. When combined with a conventional magnet, the meander coil is mainly subjected to a perpendicular magnetic field and the horizontal magnetic field is small. When combined with the new magnetic configuration, the meander coil is primarily affected by a relatively large horizontal magnetic field and is also affected by the perpendicular magnetic field. However, for coil No. 2, the perpendicular magnetic field plays a major role. The perpendicular magnetic field generated by the new magnetic configuration is larger than that generated by the conventional magnet in the area where the coils are placed. Therefore, the signal amplitude in Fig. 8(b) is slightly larger than that in Fig. 7(b). The signals in Figs. 7(a) and 8(a) are obtained by combining No. 1 coil and different magnetic configurations into EMATs. In Fig. 7(a), the A0 mode signal is weak, and the A0 mode signal of Fig. 8(a) is much larger than the S0 mode signal. Since the coil No. 1 is designed based on the phase velocity of the A0 mode, the horizontal field plays a major role in generating the A0 mode. When No. 1 coil is combined with the conventional magnet, the horizontal magnetic field in the area where the coil is located is small, so the A0 mode signal in Fig. 7(a) is very small. When coil No. 1 is combined with the new magnetic configuration, the area where the coil is located is subjected to a strong horizontal magnetic field and also a strong perpendicular magnetic field. Therefore, the signal in Fig. 8(a) has two modes, and the S0 mode is very small. In the previous numerical analysis of Fig. 3(a), it can be seen that when the driving frequency of coil No. 1 is 400 kHz, the S0 mode is suppressed and the amplitude is still small despite being affected by the strong perpendicular magnetic field. Therefore, for these two cases, experimental results and theoretical analyses are in good agreement.