Magnetic-Field-Based 3D ETREE Modelling for Multi-Frequency Eddy Current Inspection

Transcription

1 Magnetic-Field-Based 3D ETREE Modelling for Multi-Frequenc Edd Current Inspection Yu Zhang and Yong i Engineering College, Air Force Engineering Universit, ShaanXi, Xi an, 738, P.R. China octz@qq.com School of Aerospace, Xi an Jiaotong Universit, ShaanXi, Xi an, 749, P.R. China ong.li@mail.tu.edu.cn Abstract. More and more solid-state magnetic field sensors such as Hall devices are used in edd current inspection (EC) to acquire magnetic field signals. This work etends the previous analtical model, i.e. D Etended Truncated Region Eigenfunction Epansion (ETREE) of EC, and focuses on establishment of 3D ETREE of multi-frequenc edd current inspection (MFEC) on stratified conductors, while taking into account the solid-state magnetic field sensors for field quantification and rectangular coils for field ecitation. 3D Finite Element Modelling (FEM) and a hbrid modelling are adopted for verification of the established model. It has been noticed that the 3D ETREE implements the fast and accurate computation of magnetic field signals of MFEC. Following that, the directional characteristics of EC with rectangular ecitation coils are investigated, which reveals that the coil width contributes more to the measurement sensitivit than the coil length, and benefits the evaluation of defects and anisotropic conductivit profile of conductors in the follow-up stud. Kewords: Edd current inspection, Magnetic field, Etended Truncated Region Eigenfunction Epansion, Finite element modelling. Introduction To evaluate the integrit and structural health of metallic structures such as stratified conductors, Electromagnetic Non-destructive Evaluation (ENDE) techniques, especiall Edd Current (EC) and transient edd current, are preferred and used in realtime inspections []. Edd-current testing traditionall relies on the detection of impedance changes in a pickup coil as it moves across the inspected specimen. Accordingl, the theoretical modelling for EC previousl was implemented merel to predict: () the impedance signals from the stranded induction coils for time-harmonic field [], [3]; and () the electromotive force (EMF) signals from coils for transient field [4]. However, it is formidable for traditional EC to detect deep flaws in conductive materials, because low frequenc ecitation is demanded to allow edd currents to penetrate deepl into the conductors while the sensitivit of normal pickup coils, which is proportional to D. i, Y. iu, and Y. Chen (Eds.): CCTA, Part IV, IFIP AICT 347, pp. 3,. IFIP International Federation for Information Processing

2 Y. Zhang and Y. i the ecitation frequenc, is decreased. In light of this, it is more advantageous to measure the magnetic field using solid-state magnetic field sensors such as Hall sensors, giant magnetoresistive (GMR) sensors, or superconducting quantum interference devices (SQUIDs) [5], [6] are used. The theoretical modelling for EC in conunction with solid-state magnetic field sensors has been conducted for ears. Ward and Moulder proposed an epression for transient EC signals to laered conductive structures using Hall device based on the infinite integral formulation developed b Dodd and Deeds [7]. However, the sensing element in Hall devices was not taken into account. i and Tian etended the Truncated Region Eigenfunction Epansion (TREE) modelling and made it capable of predicting the magnetic field signals while the dimension of the sensing element was considered [8], [9], []. Nevertheless, the model was onl applicable to circular coils rather than rectangular coils. This paper elaborates the formulation of epressions of 3D magnetic field for Directional Edd Current Inspection (DEC) with rectangular coils via 3D Etended Truncated Region Eigenfunction Epansion modelling (3D ETREE), which concerns the geometr of rectangular coils as well as the dimension of sensing elements. The rest of the paper is organised as follows: Section presents the formulation of the closed-form epressions of magnetic field signals from solid-state magnetic field sensors; the verification of 3D ETREE b comparing with simulation results from 3D Finite Element Modelling (FEM) and a hbrid modelling is ehibited in Section 3; Section 4 elaborates the analsis of directional characteristics of DEC via investigation of the influence of the length and width of a rectangular coil on the measurement sensitivit of DEC to conductivit variation in a conductive halfspace. Theoretical Background The 3D ETREE modelling using Second-order Vector Potential (SOVP) formulation is conducted with a rectangular coil (, ) placed over the laered conductive sample which is shown in Figure. The rectangular coil, with N number of windings, is supposed to be supplied with a pulsed ecitation current, each harmonic of which is written as I(ω i ) at the angular frequenc of ω i. z h.5h z z z c c σ μ z-d z-d σ μ σ 3 μ 3 z-d 3... z-d - σ - μ - z-d - σ μ (a) Fig.. A rectangular coil and a magnetic field sensor placed over a multilaered conductor: (a) side view in -direction; (b) side view in -direction

3 Magnetic-Field-Based 3D ETREE Modelling for MFEC 3 z h.5h z z z W c c σ μ z-d z-d σ μ σ 3 μ 3 z-d 3... z-d - σ - μ - z-d - σ μ (b) Fig.. (continued). Magnetic Field at a Point of (,, z) Using SOVP formulation, the magnetic vector potential can be written as []: A W z + z W ). () ( a b Thus, magnetic field between the bottom of the rectangular coil and the upper surface of the conductor ( z z ) can be epressed as []: W B z a. () Similar to D ETREE modelling, the infinite problem region in 3D is truncated and recast with finite lengths of h and h in and directions, respectivel, which can be seen in Figure. The truncation results in the replacement of the infinite integrals with series Eigenfunction epansions. The Eigenvalues (k and k ) in two truncation directions are computed b appling the boundar condition B at z, h and h, which gives []: k nπ, h k mπ, h h. (3) h The magnetic field for each harmonic in the pulsed ecitation current is written as: ( k ) sin( k ) sin ( ) λ z z V B i e λ ω + e. (4) m n U ( ) C k + k + z

4 4 Y. Zhang and Y. i where,, and z are unit vectors. C denotes the coil coefficient for rectangular coils, which is written as: μχ C sin χ sin c λz λz ( e e ) 8 ( z z ) [( k k ) c + k k ] sin( k k ) [( k + k ) c + k + k ] sin( k + k ) ch h k ( k k NI ( ωi ) kh sin λ k ( k ) + k ) k h sin. (5) V /U is the conductor reflection coefficient in the region between the bottom of the coil and the upper surface of the conductor, which is computed b using the iterative equation: U V γ U γ d γ γ e μ μ γ γ + e μ μ γ μ λ i γ + μ ; γ γ λ + ω μ σ ; λ V ( d d ( d d k γ μ ) ) V V + k + + γ γ + + U μ μ γ + γ U μ μ γ μ + +. (6) The subscript iterates from - to.. Integral of Magnetic Field over a Sensor Volume Suppose the volume of the sensor element is W (c -c ). The location of the sensor is at the point of (,, c ), c (c +c )/. Magnetic field within the sensor element, which is placed between the bottom of the rectangular coil and the upper surface of the first laer of the conductor can be epressed in 3D as: ( ωi ) dv ( c c ) z 4 ( ωi ) ( c ) B B dddz v Bv ( ω i ). (7) 4W W c

5 Magnetic-Field-Based 3D ETREE Modelling for MFEC 5 Two tpes of sensor dimensions are considered in the model: cubiod shape and clindrical shape. For a cubiod-shaped sensor with dimension of W (c -c ), Eq. (7) is rewritten as: m n Bv i ( k + k + z ) CFG kk ( ω ). (8) W ( c c ) where, F G sin V + U ( k ) sin( k ) sin( k W ) sin( k ) e λ ( ) c c e λ c e λ λ c. (9) For clindrical-shaped sensor with dimension πr (c -c ), RW, Eq. (7) is modified as: m n Bv i ( k + k + z ) CEG kk ( ω ). () R ( c c ) where, E [ cos( k ) H ( k R) + sin( k ) J ( k R) ] [ ( ) ( ) ( ) ( )] cos k H k R + sin k J k R. () It is noted that E, F and G are the sensor coefficients, which depend on the sensor dimensions. Eqs. (8)-() give the epressions of averaged magnetic field within a sensor volume, which can be used for predicting the magnetic field signals acquired from solidstate magnetic field sensors. Following the formulation of field in frequenc domain, the discrete time-domain signal i.e. the transient EC response to stratified conductors can be recovered and epressed in the Fourier manner as [9]: π B v M M ( t ) B ( ) e M k,,,..., M i k v i π ki ω. () where, e M is a primitive M th root of unit. It is noteworth that Eq. () can be efficientl calculated b using MATAB routine ifft based on Inverse Fast Fourier Transform.

6 6 Y. Zhang and Y. i 3 Corroboration In an effort to verif the 3D ETREE modelling, Finite Element Modelling (FEM) is used to simulate the field signals from a cubiod-shaped sensor, when a rectangular coil is placed over a conductive plate. The sensor is placed at the centre of the rectangular coil and mm above the upper surface of the plate. The dimension of the sensor element is: W.9mm; c -c.5mm. The parameters of the coil are listed in Table. The conductivit and relative permeabilit of the conductive plate are 37MSm - and, respectivel. Table. Parameters of the ecitation coil Coil length / mm Coil width / mm Coil height z -z / mm Winding thickness - / mm Design ift-off c / mm Number of turns N Ecitation frequenc Current in the coil I / ma 5 Hz-kHz 5 The verification is conducted via simulations in frequenc domain. The number of elements (NOE) used in FEM is 4868 and the number of degrees of freedom (NOD) is The z-component of magnetic field is acquired and compared. The real and imaginar parts of the field signals against frequenc are compared between ETREE and FEM. The results are shown in Figure (a). The ETREE results with and without consideration of the dimension of sensor element are shown in Figure (b). The evaluation of agreement in the two comparison cases is realised b using Normalised Root Mean Square Deviation (NRMSD). The computed NRMSD values are listed in Table. (a) (b) Fig.. Comparison of the magnetic field signal (z-component) predicted b ETREE and FEM: (a) Case : ETREE considering sensor dimension vs. FEM; (b) Case : ETREE considering sensor dimension vs. ETREE predicting field at sensor point

7 Magnetic-Field-Based 3D ETREE Modelling for MFEC 7 Table. Computed NRMSD values for the two comparison cases NRMSD Case : ETREE considering sensor Case : ETREE considering sensor dimension vs. ETREE predicting dimension vs. FEM field at sensor point Real part Imaginar part Real part Imaginar part.%.7%.48%.57% It can be seen in Figure (a) and Table that the predicted signals from sensors b using 3D ETREE have good agreement with that acquired from FEM simulations, since the NRMSD values are much less than %, which indicates less residual variance between results from 3D ETREE and FEM. It is also note worth that the computation time of 3D ETREE (less than s) is much less than that of FEM (4 hours) due to large number of NOE and high NOD. Therefore, 3D ETREE can be found advantageous over FEM in terms of high computation accurac and less simulation time, which facilitates the forward modelling and inverse modelling of EC with rectangular coils. Figure (b) and Table also presents the comparison results regarding 3D ETREE with and without consider the sensor dimension in the modelling. The discrepanc is up to.5%, which indicates that the sensor dimension should be taken into account if higher modelling accurac is pursued, especiall in the case where the dimension of sensor is comparable to that of the ecitation coil. 4 Sensitivit Stud for DEC Following the verification of 3D ETREE, the sensitivit is investigated for DEC with pulsed ecitation. The maimum amplitude of the pulsed current is 5mA. The ccle and the rising time constant of the current are 5ms and μs, respectivel. The peak value (PV) against different conductivities of an isotropic conductive half-space is simulated with reference to various dimension of the rectangular coil. Thanks to the high-efficienc 3D ETREE, the database depicting the relation of PV with different dimensions of rectangular coils can be readil established. In order to obtain PV, the conductivit of the half-space varies and is written as: σ i σ (+Δσ i ), where, σ 37MSm - ; Δσ i varies from -% to %. The PVs are etracted in transient EC differential signals ΔB zi, after subtracting the individual signals B zi (when Δσ i ) into the reference signal B z(σ) (when Δσ i ). For a particular rectangular coil with the length 9.5mm and the width 6mm, the PV against Δσ i is presented in Figure 3(a). Figure 3(b) shows the plot with errorbars indicating the variation in the curve when () the coil length is increased to.45mm; () the coil width is increase to 6.6mm. It can be seen in Figure 3(a) that the curve of PV ehibits good linearit when Δσ i varies in the small region from -% to %. The slope of the curve (k ab dpv/dδσ i ), which is -. in Figure 3(a) indicates the measurement sensitivit to the variation in the sample conductivit. High value of curve slope implies that high measurement sensitivit, which gives significant variation in PV due to the change in conductivit. The curve slope is found varing with the coil dimension, which can be seen in Figure 3(b).

8 8 Y. Zhang and Y. i (a) (b) Fig. 3. PV against Δσ i for (a) the coil (9.5mm and 6mm) and (b) coils (9.5mm and 6mm;.45mm and 6mm; 9.5mm and 6.6mm) The measurement sensitivities against different combinations of and of the coil (6mm 9.5mm) are modelled. In order to investigate the influences of and on the measurement sensitivit, surface fitting using a quadratic function is implemented after the database (k ab vs. coil dimension) is built up. The results are presented in Figure 4. Fig. 4. The measurement sensitivit vs. coil dimension The quadratic fitting function is written as: 4 ( ) + ( ) k ab. (3) The fitted curve has 98% agreement with the database, which is acceptable for the investigation. The influence of and on the sensitivit k ab can subsequentl be analsed via taking first-order derivative of k ab with respect to as well as, which can be written as: κ ( k ab ) and κ ( k ab ). Since the magnitudes of κ and κ are of interest, the absolute values of κ and κ are computed and shown in Figure 5.

9 Magnetic-Field-Based 3D ETREE Modelling for MFEC 9 Fig. 5. Influence of and on the sensitivit k ab : plots of κ ( ) and κ ( ) k ab k ab It can be found in Figure that the width () plas more important role than the length () regarding the contribution to the measurement sensitivit to conductivit variation. The reason lies in the fact that the distance between edd currents flowing oppositel under the two lengths of the coil varies with the coil width. Small width results in more cancellation of edd currents. Since edd currents induce the secondar magnetic field, which is depends on the conductivit of the sample, the cancellation of edd currents causes decrease of measurement sensitivit. As can be seen in Figure 5, when the width increases, the rate of enhancement of measurement sensitivit rises, which is because less edd currents are cancelled out. Although the increase in coil length enhances the measurement sensitivit due to increased distance between edd currents flowing oppositel under the two coil widths, the enhancement due to the coil length is less than that due to the coil width. This is because of the fact that the length is larger than the width, and thus the edd current densit under the coil width is higher than that under the coil length. Therefore, the dependenc of measurement sensitivit on the coil length is less than that on the coil width. Compared with circular coils whose sensitivit is dependent on the coil diameter, the rectangular coils show the distinct characteristics of measurement sensitivit, which is mostl influenced b the coil width. The contribution to the sensitivit from the coil width is more than that from the coil length. As a result, EC with a rectangular coil shows the directional measurement sensitivit, which is along the width of the coil. In such case, DEC is realised, which is advantageous over traditional EC in terms of () high detectabilit of cracks with different orientations; () better characterisation of anisotropic distribution of conductivit due to applied stress or strain in particular directions. 5 Conclusion This paper presents the 3D ETREE modelling for prediction of magnetic field signals from solid-state magnetic field sensors, which can be placed at an arbitrar location between bottom of the ecitation and the upper surface of the conductor. The consideration of sensor dimension in the formulation results in the new coefficient in the

10 3 Y. Zhang and Y. i epression. The so-called sensor coefficient is dependent on the sensor geometr, and indispensable when () EC with utilisation of magnetic field sensor arras is simulated; () the sensor dimension is comparable to coil size. 3D ETREE facilitates the investigation of directional characteristics of measurement sensitivit i.e. change in PV due to conductivit variation in samples for DEC. The influences of the coil width and length on the DEC sensitivit is analsed b setting up a database depicting the relation between measurement sensitivit and coil dimension. It can be noticed that the contributions from the length () and the width () to the measurement sensitivit are not identical. The coil width is dominant in enhancement of measurement sensitivit. Therefore, compared with traditional EC with circular coils, which has the measurement sensitivit equall on the circumference of the probes, DEC is found having the dominant measurement sensitivit along the coil width, which benefits the evaluation and characterisation of natural cracks and anisotropic conductivit due to stress/strain. The eperimental work on verification of 3D ETREE and the directional characteristics of DEC, which has been found in theoretical stud, is currentl underwa. Further work would be focused on the identification, characterisation and reconstruction of natural defects and inhomogeneous conductivit profiles of conductors. Acknowledgement. The authors would like to thank Miss. Xinhua i and Prof. Gui Yun Tian of Newcastle Universit, UK for 3D FEM simulation and valuable discussions. References. Sundararaghavan, V., Balasubramaniam, K., Babu, N.R., Raesh, N.: A multi-frequenc edd current inversion method for characterizing conductivit gradients on water et peened components. NDT&E Int. 38, (5). Theodoulidis, T.P., Kriezis, E.E.: Edd Current Canonical Problems (with Applications to Nondestructive Evaluation). TechScience Press (6) 3. Theodoulidis, T.P., Bowler, J.R.: Edd current interaction with a right-angled conductive wedge. Proc. of the Roal Societ of ondon A 46, (5) 4. Yang, H.C., Tai, C.C.: Pulsed edd-current measurement of a conducting coating on a magnetic metal plate. Mea. Sci. & Tech. 3, () 5. Tian, G.Y., Sophian, A.: Stud of magnetic sensors for pulsed edd current techniques. Insight. 47, 77 8 (5) 6. Tian, G.Y., Sophian, A., Talor, D., Rudlin, J.: Multiple sensors on pulsed edd-current detection for 3D subsurface crack assessment. IEEE Sensors Journal 5, 9 96 (5) 7. Ward, W.W., Moulder, J.C.: ow frequenc, pulsed edd currents for deep penetration. Rev. of Prog. in QNDE 7, 9 98 (998) 8. i, Y., Theodoulidis, T.P., Tian, G.Y.: Magnetic field-based edd current modelling for multilaered specimen characterization. IEEE Trans. on Mag. 43, 4 45 (7) 9. i, Y., Tian, G.Y., Simm, A.: Fast analtical modelling for pulsed edd current evaluation. NDT & E Int. 4(6), (8). Tian, G.Y., i, Y., Mandache, C.: Stud of lift-off invariance for pulsed edd-current signals. IEEE Trans. on Mag. 45, 84 9 (9)