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This paper will discuss axial MFL applied to high-speed inspection on pipeline/tube examinations, based on numerical simulations. The MFL signals dependent on such factors as defect depth are investigated while taking probe movement at high speed into account. The organisation of the paper is arranged as follows: Section 2 gives the general overview of theoretical background on eddy currents due to probe movement and the modified Maxwell equation. Section 3 presents the numerical simulation on a high-speed MFL inspection system. Eddy currents in steel specimen are observed and their effect on MFL signals is discussed. Moreover, the MFL signals against defect depth is investigated and analysed. Section 4 introduces a proposed high-speed MFL inspection system with specifically designed layout of sensor arrays for high-speed evaluation on pipelines, tubes and rail tracks. ipelines, tubes and rail tracks.
2. Theoretical background
In electromagnetic theory, the magnetic fields under the effect of mechanical motion is governed by Lenz’ Law and Lorentz’ Law. Lorentz’ Law can be used for analysis of the moving probe effect in a dynamic MFL inspection system. If the probe moves at certain speed, the Lorentz force induces currents in conductive specimen. Such currents in specimen can be regarded as eddy currents dependent on the velocity at which the probe travels and the current density is expressed in Eq. (1) [2]:
J_v = σ v_c x Β   (1)
where, J_v denotes eddy current density in specimen; v_c denotes the speed of the applied magnetic field with magnetic flux density, B relative to the specimen. s represents conductivity of the specimen.
Since electromagnetic phenomena are governed by Maxwell’s equations in differential or integral form, eddy currents generated by moving magnetic fiels must be considered in dynamic electromagnetic systems by making modification on traditional equations with Eq. (1). With respect to dynamic EM system, the governing equation deduced from Maxwell’s equations is added with eddy currents due to movement of applied magnetic field by taking the Eq. (1) into account. The modified equation for time-harmonic electromagnetic field is expressed in Eq. (2) [9,10]:
     (2)
where, A denotes magnetic vector potential ; σ(jωA)=J_e denotes the density of eddy currents generated by applied magnetic field; Js denotes source current density; v denotes media velocity; V_loop denotes loop potential. In this equation,  stands for eddy current density related to motion of applied magnetic field.
Compared with the governing equation that precludes J_v, the modified equation implies that the eddy currents generated by moving magnetic field influence not only the currents distribution in conductive specimen but also the magnetic field profile, which results in distortion of the measured signals.
In order to investigate the EM field in systems on a macroscopic level, the governing equations are solved by using numerical methods, e.g. finite difference method (FDM) [11], finite element method (FEM) [12,13], boundary element method (BEM) [14], hybrid method with finite element method and boundary element method (FEM-BEM) [15,16], meshless method [17,18]. Subsequently, many commercial numerical simulation packages such as ANSOFT [19] and FEMLAB [20] which implement FEM whilst allowing BEM codes are programmed to simulate electromagnetic problems with moving parts by taking the eddy currents caused by movement into account and prevalent in EM NDE [21,22]. In this paper, we will apply ANSOFT Maxwell EM for simulation of magnetic field distributions and their variations under high speed movement. The work can be exploited for practical system design, signal processing and characterisation on interaction of system with defects.