NDT.net • Apr 2005 • Vol. 10 No.4 
Ultrasonic Field Modelling for Arbitrary NonContact Transducer SourceX Jian, Steve Dixon, Rachel S. EdwardsUniversity of Warwick, Department of Physics Coventry CV4 7AL, UK Phone +44 24 76574116 Fax +44 2476692016 Corresponding Author Contact: X Jian, Email: x.jian@warwick.ac.uk, Internet: www.warwick.ac.uk
© SPIE  The International Society of Optical Engineers. ABSTRACTWe present a model for transient ultrasonic wave generation by Electromagnetic Acoustical Transducers (EMATs). Analytical solutions are currently available only for few kinds of sources and our model combines these analytical solutions and numerical computation to predict the ultrasonic field generated by arbitrary sources. This model can be used to calculate bulk waves within samples as well as surface waves on sample surfaces with the advantages of explicit physical meaning and quick processing speed over pure numerical calculations such as the Finite Element Method (FEM). We use the model to explain how static and dynamic magnetic fields generate ultrasonic waves in a sample. We wish to characterize the EMAT source in detail in order to tailor sources for optimal configuration for specific NDE applications. A Michelson laser interferometer is used to measure out of plane surface displacement of sample, and results agree well with the modelling simulation. The modelling can be used for arbitrary source.1. INTRODUCTIONElectroMagnetic Acoustic Transducers (EMATs) can generate or detect ultrasound in electrically conductive or magnetic materials through the Lorentz force principle or by magnetoelastic effects [110]. In this case we have limited ourselves to investigating nonmagnetic metals where operation of the EMAT is due to Lorentz force mechanisms alone. We also consider wideband generation where the EMAT is driven by a broadband pulse of current, rather than a single frequency or pseudo single frequency (toneburst) pulse. When current is pulsed through the EMAT coil, eddy (or mirror) currents are induced in the sample skin depth. The electrons that make up this eddy current experience a Lorentz force under interaction with the dynamic magnetic field from the coil itself and if present, also with the external static magnetic field.Due to the complexity of the problem, an analytical solution is only available for some simple uniform force distributions, such as uniform normal piston force and uniform radially acting ring force [1,3, 1112]. Accordingly, transient field studies were carried out generally by the Finite Element Method (FEM) [79]. In much of the previous research, the Lorentz force contribution due to the dynamic magnetic field from the coil itself has been ignored [89]. The disadvantage of using a totally numerical, single stage calculation to obtain the ultrasonic field generated by an EMAT is that it lacks explicit physical interpretation [12, 89] of the nature of the source and the results ultrasonic field. The approach described in this paper employs FEM to calculate the transient force distribution generated by an EMAT [510] and then decomposition of the force into many force components for which analytical solutions exist. Finally the sum of the displacements from each of these components is calculated to obtain the total displacement field [1, 3]. A Michelson laser interferometer is used to measure the absolute outofplane displacement [1, 3, 6] and good agreement between the calculation and experiment is observed. Since our analysis is based on analytical solutions, the paper gives a good interpretation of the contribution to ultrasonic generation of the Lorentz force due to dynamic and static magnetic field, explains our previous novel measurements, and help to design EMAT optimally. 2. METHODFigure 1 shows the experimental arrangement of a spiral coil, where B_{s} is static magnetic field, J_{0} and J_{e} are excitation and eddy current respectively. F_{d} and F_{s} are Lorentz forces due to dynamic and static magnetic fields respectively.The induced eddy current amplitude decreases exponentially with depth, and exists mainly within the skindepth. In this paper, the excitation current has frequency around 300 KHz. We use the force distributions along the X direction at depth 0.01mm to calculate the ultrasonic displacement waveforms. As a example, figure 2 shows how the spatial outofplane Lorentz force due to dynamic magnetic field generated by the spiral coil is decomposed into six uniform normal piston forces schematically. In fact, the number of decomposed forces is bigger than 6 and is dependent on the spatial distribution of Lorentz force.
Assuming the Lorentz force is decomposed spatially into N force components, the total displacement for these temporal delta force components is given by,
where denotes a convolution operation, is the temporal dependence of Lorentz force, is the total displacement for these temporal delta force components of each circular source (piston or ring source), and is given by,
where r_{i} is the radius of uniform piston or ring force components and y_{max} is the depth in metal sample where maximum Lorentz force is observed. _{} is ultrasonic wave generated by force component i with temporal dependence of delta function [1112]. In same way, the solution to line coil can be deduced, but will not be given here due to limit space. 3. RESULTS3.1 CalculationA spiral coil and a line coil made from insulated copper wires of radius 0.15 mm are considered for Rayleigh wave generation. The spiral coil is of radius of 5 mm. The liftoff between coil and sample is 0.1 mm. The sample is a thick aluminium block, and static magnetic field is 0.395 Tesla. The current direction of the spiral coil and the line coil is indicated in figure 1. We use Finite Element Method to calculate the Lorentz forces in the sample due to dynamic magnetic field and static magnetic field. The Lorentz forces at below sample surface 0.01 mm are taken and plotted in figure 3.
The calculated ultrasonic surface waves of the spiral coil and the line coil are shown in figure 4 and 5 respectively. We find that the direction of static magnetic field has an influence on the shape and amplitude of surface displacement generated. It should be careful to arrange the direction of static magnetic field and that of current in order to obtain an optimal surface displacement.
3.2 MeasurementA spiral coil and a line coil mentioned above are fabricated by hand. The two coils are used to generate surface waves at the liftoff of 0.1 mm. A Michelson laser interferometer that is at the same side of acting coil is used to measure the outofplane surface displacement. The advantage of using the interferometer is that the result is independent of receiver and can be scaled to true displacement [1, 3, 510].Measured results are shown in figures 6 and 7. Figure 6 shows the measured ultrasonic displacement at points x1 generated by the spiral coil, and should be compared to the simulated displacements shown in figures 4. It can be seen that there is good agreement between the measured and predicted displacements of the Rayleigh wave. Figure 7 shows the measured ultrasonic displacement at point x1, and agrees well with the simulation in figure 5.
4. DISCUSSION AND CONCLUSIONA model, which can be used to predict Rayleigh and surface skimming compression waves generated by an EMAT, has been discussed. The model combines analytical solutions and numerical calculations to obtain the displacement field for any EMAT configuration. Calculations are particularly time efficient where the source is a line or circularly symmetric source (spiral or circular coil). Two kinds of coil configurations, the line and the flat spiral or 'pancake' coil, have been calculated as examples. Calculation results agree well with the measurements by a Michelson laser interferometer.The induced eddy current and dynamic magnetic field are independent of the external static magnetic field. The Lorentz force in the sample surface is the cross product of the eddy current and the magnetic field (dynamic and static magnetic field). When an excitation current pulse is of duration 2.5 µs, the outofplane Lorentz force due to the dynamic magnetic field is about 10 times larger than the inplane Lorentz force, and the force due to the dynamic magnetic field is about 5 times larger than the Lorentz force due to the external static magnetic of intensity of 0.395 Tesla. The Lorentz force due to the dynamic magnetic field always acts as a repulsive force between the EMAT coil and sample. Concepts of the dynamic magnetic field, the Lorentz force due to the dynamic and static magnetic fields and separated outofplane and inplane Lorentz forces have been found to be very helpful in explaining the EMAT mechanisms for ultrasonic generation. Two examples given here of typical EMAT coils have demonstrated the usefulness of the model in ultrasonic field prediction, and have already explained some novel experimental measurement results. It will certainly help with EMAT design to obtain an optimal ultrasonic field of desired pulse shape and maximum efficiency. Although only Rayleigh waves on the sample surface have been considered here, this model could be used to give displacement prediction at a field point below the sample surface, which will be discussed in another paper. ACKNOWLEDGMENTSThe work is supported by the EPSRC funded UK Research Centre for NonDestructive Evaluation (RCNDE). The authors would thank Mr John Reed and Dr Mark Potter for their valuable help with the job.REFERENCES

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