On the modelling of a Barkhausen sensor

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1 On the modelling of a Barkhausen sensor More info about this article: Abstract Gert Persson Chalmers University of Technology, Sweden, gert.persson@chalmers.se A numerical model of a plain Barkhausen sensor is studied by means of the finite element method. The three-dimensional sensor consists of a magnetizing yoke, modelled as an C- core electromagnet. The planar test object is modelled as a thin plate consisting of two or three layers, indicating a situation with a hardened layer, a transition zone and a soft layer. Simulations for the magnetic field are made for different frequencies and material combinations in the frequency plane, corresponding to a harmonic excitation of the electromagnet. All materials, including the C-core, are modelled as magnetic isotropic having a linear magnetic permeability in the constitutive relation. The magnetic flux density is computed on both the surfaces of the planar test object and as a function of depth. The effect of lift-off is studied. 1. Introduction Non-destructive testing (NDT) is used in an increasingly degree in industry. The purpose may be to ensure safety or service life (In Service Inspection) or to ensure quality in manufacturing. In both cases, it is about detecting, as well as the size and positioning of defects, or indirectly determining material properties with non-destructive methods. One of these NDT methods is commonly referred to as Magnetic Barkhausen Noise (MBN). MBN is a method based on measurements of response to changes in a periodically varying magnetic field. Simplified reasoning, the measuring device is based on a magnetic yoke which generates a low frequency magnetic field and a coil which via induction picks up the response from the sample. The measured response varies greatly in time when the magnetic domains in the material are first magnetized in one direction and then reversed. This local magnetization of the domains occurs abruptly and has given rise to the term "Noise" or Barkhausen effect. Apart from the stress state, microstructure and any defects, the detected Barkhausen signal also depends on the generated magnetising field. The field is produced by the sensor and is then dependent on parameters such as coils, core geometry, sensor to specimen coupling (lift-off) and core material. Several Finite Element (FE) studies have been made on the parameters of the yoke (1, 2, 3) In the current study the computations are made for the full 3D case in the frequency domain, and with a focus on the layered sample. One important parameter for judging the performance of a sensor is the depth of penetration. This can be calculated by using standard formulas, which here is compared with numerical estimations. Creative Commons CC-BY-NC licence

2 2. Theory 2.1 Magnetic field By analysing the results of the magnetic field computations, some estimations about the depth and location of the Barkhausen noise signal can be made. In this study the magnetic field, used to excite the Barkhausen emission in a sample, is computed in terms of magnetic flux. This approach is commonly used (1) for studying performance and optimizing sensor devices. There is then no need for any special model for the emission. The resulting magnetic flux will be computed for the case of harmonic excitation of the magnetic device. The motivation for this approach is the following The resulting flux density distribution B in the sample material is described by: B = H (1) where is the permeability of free space and the permeability of the material. The applied excitation magnetic field H, that is the magnetic field strength in the core, follows from Amperes circuit law: H = (2) where is the number of turns in the coil, is the coil current and is the mean length of the magnetic flux path in the circuit. In line with a previous study (4) Faradays law can be used to show that the voltage induced in a pickup coil will be proportional to the average flux density B in the coil. The detected MBN can then be considered proportional to the resulting flux density B and there is then no need to model a pickup coil. This approach does not consider for the special Barkhausen effects, but could be used for predicting the average MBN behaviour. 2.2 Depth of penetration Electromagnetic signals pass through a conductive material and attenuate by eddy current damping. The standard depth of penetration (SDP), the value δ where the magnetic field has decreased to e -1 or 36.8 % from the value at the surface, is calculated as = (3) 0 where σ is the conductivity, µr is the relative permeability, µ0 is the permeability in vacuum and f is the frequency. The expression for the skin depth is derived for a plane wave impinging on a flat surface and the actual penetration depth in eddy current experiments strongly depends on the instrument, the probe, the environment and on the defect detected (5). According to (5) the frequency is often set to three SDP (3δ) within the sample when eddy current is used to measure the electrical properties, such as conductivity of a material. The material thickness will then not affect the measurements since the eddy currents will be so weak at the back side of the material. 2

3 3. Model The software COMSOL Multiphysics were used for making a finite element model of the magnetising unit and the structural layers. The magnetic flux densities were computed for different parameter settings of both the magnetising yoke and the different layers of the sample. The numerical computations were made in the frequency-plane for two different frequencies, corresponding to two different harmonic excitations in the time-plane. The model includes magnetic field and induced current distributions in and around the core and coil. The numerical 3D finite element model consisted of a magnetising unit and a flat sample, surrounded by air, as shown in Fig. 1. The magnetising unit was designed as a typical Barkhausen sensor (4) consisting of a soft iron C-core and a cupper coil. The properties of the unit are listed in table 1. Figure 1. The magnetising unit, layered sample and coordinate system. Figur 1 shows the geometry of the magnetising unit and the layered sample. The dimensions of the circular core is L h and the lift-off is g. The total thickness of the layers is t. The thickness of the uppermost layer is tm and the middle layer tb. Table 1. Properties of the magnetising yoke Magnetising Voltage Number of coil turns L h t Diameter of core 10 V mm 26 mm 5 mm 18 mm When it comes to modelling the tested sample, a layered structure is chosen according to findings in (6). It is observed that for an induction hardened carbon steel, micrographs for the cross-section shows a distinct layered structure, consisting of a 3 mm top layer of martensite, a middle transition layer of martensite/bainite and bainite/martensite mixture around 1.5 mm. Below 5 mm no further layered structure is detected in the perlite and ferrite bulk material. From these findings, it is chosen to model a layered structure consisting of an upper layer and a lower layer with material parameters according to table 2. The total thickness of the flat sample is in all cases 5 mm and two different structures are studied. The first consists of an upper layer of martensite and a lower of bainite, thickness tm and tb. The second consists of three layers, in order, martensite, bainite and pearlite. The sample is surrounded by air and the magnetizing yoke is placed on the surface of the upper layer, see figure 1. The material properties were chosen in accordance with the results from 3

4 JMatPro simulations (6) and is presented, together with the SDP calculated according to formula 3, in table 2. Table 2. Properties of the different microstructures Martensite Bainite Pearlite Relative magnetic permeability Electrical conductivity (S/m) 6.40 x x x 10 6 SDP (mm) at 125 Hz SDP (mm) at 50 Hz The finite element model contains of approximately elements, yielding a problem of degrees of freedom (dof). The model is made large enough for making boundary effects vanish. For current flow computations, the Coil Geometry Analysis study step were used. Different configurations of elements were chosen dependent of frequency and layer thickness. Some of the computations were made using magnetic symmetry boundary conditions. 4. Results and discussion Figure 2. Magnetic flux density on the top surface (z = 0). Two layers structure, frequency 125 Hz Figure 2 shows the x-component of the magnetic flux density as a function of the x- coordinate on the top surface (z = 0) for 125 Hz. The layer consists of martensite on top of bainite, martensite having the thickness tm of 0, 1, 2, 3 and 4 mm. Total layer thickness is 5 mm. The case of no martensite, tm = 0, corresponds to pure bainite. There is a difference in the flux between the yoke legs. The signal for 1 mm martensite is approx. 9% larger than the signal for 2 mm and 15% larger than for 3 mm. There seems to be no difference between martensite layers thicker than 3 mm. Pure bainite gives a flux that is 68% larger than pure martensite. Furthermore, the area between the yoke legs seems to be the obvious place to put a pickup coil 4

5 a, b, c, d, Figure 3. Magnetic flux density contour, arrows showing magnetic flux density in plane Two layers structure, frequency 125 Hz a, tm = 1 mm b, tm = 2 mm c, tm = 3 mm d, tm = 4 mm Figure 3 shows the magnetic flux density in the xz-plane (y = 0) for positive x- coordinates, close to the right yoke leg, 125 Hz. The black arrows correspond to proportional magnetic flux density. The layered material matches to the one in figure 2 with a martensitic layer on top of bainite, martensite having the thickness tm. The magnetic flux density is strongly affected by the layered structure. The flux density on the surface, right under the centre of the yoke is in all cases approx T and approx T on the opposite side of the sample. For the case of martensite thickness tm = 1 and 2 mm the magnetic flux clearly follows the layered structure. This is less obvious for tm = 3 mm and almost gone for the thickness of 4 mm. This also corresponds to the flux computed on the surface in figure 2, there is almost no difference in the response between martensite of thickness 3 and 4 mm. The SDP calculated by formula 3, and shown in table 2, is 1.56 mm for martensite at 125 Hz. It is seen in figure 3 that when tm is of the same order as the SDP the structure still 5

6 affects the magnetic flux, and then contributes to a magnetic response that will affect the Barkhausen noise. Figure 4. Magnetic flux density on the opposite surface (z = -5mm). Two layers structure, frequency 50 and 125 Hz Figure 4 corresponds to figure 2 and shows the x-component of the magnetic flux density as a function of the x-coordinate, but here on the opposite side of the sample (z = -5 mm) for 50 and 125 Hz. Figure 4 shows that the flux density for 125 Hz is less than 5% of the value in fig 2 (on the top surface of the sample) and could possibly be hard to detect experimentally. For the given yoke parameters, the flux for the 50 Hz on the opposite side is of the same order as the flux for the 125 Hz on the top surface in fig 2. The SDP according to table 1 shows that 3δ is 4.65 mm for martensite and 3.15 mm for bainite at 125 Hz. Since the thickness of the sample is 5 mm it is expected that 125 Hz yields a weak flux. For the case of 50 Hz the sample is almost within 2δ (4.92 mm) for martensite and 3δ (5.13 mm) for bainite. The weakest signal peak (tm = 0) for 50 Hz is approx. 30% of the strongest signal (tm = 4). In figure 2 and 3 it is numerically shown that it is possible to tell the difference between a layered and an unlayered structure if the martensite thickness is less than 3 mm for 125 Hz. For a structure of 3 layers one can assume that all three layers should be within this limit to be possible to detect. This is numerically investigated in figure 5. 6

7 a, b, c, d, Figure 5. Magnetic flux density contour. Three layers structure Frequency 50 Hz: a, tm = tb = 1.5 mm b, tm = tb = 2 mm Frequency 125 Hz: c, tmv= tb = 1.5 mm d, tm = tb = 2 mm Figure 5 is like figure 3 and shows the magnetic flux density the xz-plane (y = 0) for 50 and 125 Hz. The thickness of the two upper layers tm and tb, consisting of martensite and bainite, is here mutually 1.5 and 2.0 mm. The remaining sample thickness, the third layer, consists of perlite according to table 2. The SDP in martensite from table 2 is 5 = 2.46 and 5 = 1.56 mm for 50 and 125 Hz, respectively. Figure 5c shows that for tm = tb = 1.5 mm < 5 the magnetic flux seems to penetrate deeper than one SDP = 5. Figure 5d, where tm = tb = 2 mm > 5, indicates that the magnetic flux follows the formula 3. For the 50Hz case the flux in figure 5b, tm = tb = 2 mm,seems to affect deeper layers than in 5a where tm = tb = 1.5 mm. In both cases tm = tb < 5. It seems like when the thickness of the upper martensitic layer is greater than the corresponding SDP, the flux follows formula 3. When the martensitic layer is less than the SDP the flux has a larger effect on the following layers and this has to be investigated further. 7

8 Figure 6. Magnetic flux density on the top surface (z = 0). Two layers structure, martensite 2mm, frequency 125 Hz, different lift-off. Figure 6 corresponds to figure 2 in the sense that it shows the x-component of the magnetic flux density as a function of the x-coordinate on the top surface (z = 0) for 125 Hz. Here the layers consist of 2 mm martensite and thereby 3 mm bainite. The effect of lift-off g is computed for 0, 0.5, 1 and 1.5 mm. The obvious result is that the edge effects from the yoke disappears when the lift-off is nonzero. The difference between no lift-off and 0.5 mm lift-off is a reduction of approx. 15%. A lift-off of 1 and 1.5 mm yields a reduction of 21% and 25% respectively. The effect of 0.5 mm lift-off, on the flux density on the surface at 125 Hz in figure 6, is of the same order as 1.0 mm increase of the martensitic layer in figure Conclusions From the numerical computations, for the given probe parameters and sample geometry, the following conclusions can be drawn: It is possible to see the difference in the magnetic flux on the yoke surface for the case of upper martensite layer thickness of 0, 1, 2 and 3 mm. The remaining 5 mm sample consists of bainite. For the case of 4 mm martensite layer this cannot be distinguished from the 3 mm case. The magnetic flux on the backside of the sample is less than 4% of the one on the yoke side for the 125 Hz computation. This is consistent with the statement that three SDP makes the material thickness large enough for not affecting the measurements. 8

9 It is seen that when the martensite thickness is of the same order as the SDP the structure still affects the magnetic flux, and then contributes to a magnetic response that will affect the Barkhausen noise. When the martensitic thickness is less than the corresponding SDP the flux can have a larger effect on the following layers than the case of pure martensite. The response from these layers will affect the Barkhausen noise and this has to be further investigated. For the case when the martensitic layer is less than one SDP the effect of 0.5 mm lift-off, on the flux density on the surface, is of the same order as 1.0 mm thickness increase of the martensitic layer. Acknowledgements Swedish s Innovation Agency (VINNOVA) is gratefully acknowledged for the financial support in the Non-destructive Characterization Concepts for Production project FFI- OFP4P.WP3 References 1 N Prabhu Gaunkara, O Kypris, I C Nlebedim, and D C Jiles, Optimization of sensor design for Barkhausen noise measurement using finite element analysis, J. Appl. Phys., 115, 17E512, M Katoh, K Nishio, T Yamaguchi, FEM study on the influence of air gap and specimen thickness on the detectability of flaw in the yoke method. NDT & E Int,Vol. 33, Issue 5, pp , 3 J Paľa, J Bydžovský, O Stupakov, I Tomáš, J Kadlecová and V Jančárik, Influence of yoke legs shape on air gap uncertainty, J. El. Eng., Vol. 57, No. 8/S, pp , S White, T Krause and L Clapham, Control of flux in magnetic circuits for Barkhausen noise measurements, Meas. Sci. Tech., Vol. 18, pp , G Mook, O Hesse and V Uchanin, Deep Penetrating Eddy Currents and Probes. Materials Testing ECNDT 2006 Conf. Proc. Tu.3.6.2, Berlin, E Tam, G Persson, P Hammersberg, A Stormvinter and J Olavison, Preliminary study: Barkhausen noise evaluation on the Hardening Depth of Induction-hardened carbon steel, 12 th ICBM Conf. Proc. pp 73-83,