A method of determining eddy current instrument response from models S Majidnia a,b, C Schneider b, J Rudlin b,r Nilavalan a a Brunel University, Kingston Lane, London,UB8 3PH b TWI, Granta Park,Cambridge CB21 6AL, UK Telephone: 01223 899000 Fax: 01223 890689 E-mail: shiva.majidnia@brunel.ac.uk Abstract The output of a model of an eddy current test is typically the impedance change of a coil or probe. An eddy current instrument takes this impedance change and by means of the electronic circuits and instrument settings, modifies it to produce a screen display suitable for the operator. To predict the actual output of an eddy current test is, it is important to know the effect the instrument has on the impedance change as output by a model. By using the output of standard calibration situations, and using a mathematical transformation to estimate the instrument output equivalent to the impedance change produced in the models, the output of an instrument to a more general modelled situation can be estimated. This paper describes the method developed and gives some practical examples. 1. Introduction It is desirable to be able to model NDT tests, to avoid the necessity of extensive experimental programs. Various modelling software are available to investigate the eddy current testing using finite element methods (FE) such as CIVA, AMPERES and COMSOL Multiphysics. These are now capable of modelling the electromagnetic fields from coils, and a wide range of flaw situations in 3D (1). The model can provide various outputs which can be chosen according to need of the project. The Impedance of the coil for different situations is one of the most common outputs that the software can produce. However this does not take into account the instrumentation and display of the eddy current data. This paper describes the 3D FE modelling of an eddy current NDT technique, which predicts the impedance of a coil for various thicknesses of a plate and also for EDM slots introduced into the plate. A transformation of the coil impedance obtained from the models to the instrument output of an Ether NDE Veritor eddy current detector instrument is then established for the thickness change in an Inconel plate (as a means of calibration). The transformation is then used to predict the instrument output for the slots. The prediction is then compared with measurement. The purpose of this work is to enable the output of an instrument to be predicted for any 3D flaw situation.
1.1 The finite element modelling The first step in using the finite element analysis (FEA) is to describe the problem. It is often necessary to simplify a real engineering problem into an idealised problem that can be solved by FEA. After specifying the physical model, the next step was to discretize/mesh the solid. (2).This was the starting point of the finite element method. Meshing partitions the geometry into small units of a simple shape called mesh elements. The mesh generator discretizes the domains (intervals) into smaller intervals (or mesh elements). The governing equations of the appropriate physics are then discretized and solved inside each of these sub-domains. The finite in FEA, refers to the limited number of elements used to represent the structure. The elements are of finite, measurable size. A computer needs to be able to handle the computations for this finite number of elements. Each element acts on its neighbouring elements. FEA assembles the equations from all the elements into one large matrix equation, and the computer is used to determine the numerical solution. A key concept of FEA is this: if the elements are made small enough and are spread advantageously across the part, the numerical solution can closely approximate reality. 2. Eddy current modelling using COMSOL Multiphysics In order to model the behaviour of an eddy current system as applied for non-destructive testing, the magnetic field interface of AC/DC module of COMSOL Multiphysics was used. The magnetic fields interface has the equations and external currents for modelling magnetic fields and solving for the magnetic vector potential. The main feature is the Ampère s Law feature, which adds the equation for the magnetic vector potential and provides an interface for defining the constitutive relation and its associated properties such as the relative permeability. 2.1 Field equations For general time-varying electromagnetic fields, Maxwell s equations are (3) : (1) (2) (3) (4) The first two equations are also referred to as Maxwell-Ampère s law and Faraday s law, respectively. The last two are forms of Gauss law in the electric and magnetic 2
form, respectively. The notation used in these (and subsequent) equations are defined in Table 1. To obtain a closed system, the constitutive relationships describing the macroscopic properties of the medium are also needed, as described below (4) : (5) ( ) (6) The electric polarization vector, P, is generally a function of E and describes how a material is polarized when an electric field, E, is present. Some materials can have a non-zero P in the absence of an electric field. The magnetization vector, M, is a function of H, and similarly describes how a material is magnetized when a magnetic field, H, is present. One use of the magnetization vector is to describe permanent magnets, which have a non-zero M when no magnetic field is present. For linear materials, the polarization is directly proportional to the electric field and magnetization is directly proportional to the magnetic field. For nonlinear materials, a generalized form of the constitutive relationships should be considered. This relationship is expressed as below: (7) (8) (9) Where ε r is the material s relative permittivity and μ r is the relative permeability. D r is the remnant displacement which is the displacement when no electric field is present and B r is the remnant magnetic flux density, which is the magnetic flux density when there is no magnetic flux present. 3
Table 1. Notation Symbol Quantity Symbol Quantity E Electric field V Electric potential intensity H Magnetic field intensity M Magnetization vector D Electric flux density r Electric charge density B Magnetic flux ε Electric permittivity density J Current density μ Magnetic permeability A Magnetic vector potential σ Electric conductivity 2.2 The model For the modelling in this paper an air coil probe was used as the exact properties of a ferrite core were not available. A 3D modelling approach was used to enable the introduction of slots at a later stage. When modelling a multi-turn coil in COMSOL Multiphysics, it is not necessary to model the connecting leads and individual wires. The software models a homogenized current carrying region to compute the magnitude and direction of the current flow. It then uses this information to find the magnetic field in and around the conductor. This method is very useful when modelling a large number of turns, as each individual turn is also isolated, so there is no short-circuiting between conductors (Figure 1). Figure 1. Typical model of a multi-turn coil In order for the model to represent a real scenario, an air domain needed to be used around the conductor. A magnetic insulation boundary was then used on the boundaries of the air domain, which sets the magnetic potential to 0 at the boundary (5) : A z = 0 (10) 4
The air domain that is modelled with finite elements is unbounded and open in real scenarios, meaning that the electromagnetic field extends towards infinity. The easiest approach to modelling an unbounded domain is to extend the simulation domain far enough that the influence of the terminating boundary conditions at the far end becomes negligible. This approach can create unnecessary mesh elements and make the geometry difficult to mesh due to large differences between the largest and smallest object. To resolve this problem an infinite element domain was used. (6).This implementation mapped the model coordinates from the local, finite-sized domain to a stretched domain. The inner boundary of this stretched domain coincided with the local domain, but at the exterior boundary the coordinates were scaled towards infinity. This approach was not computationally expensive and made the model more physically feasible. Table 2 contains the dimensions used for both the modelling and the experimental coil for this paper. Table 2 Coil dimensions for the experiment and the model Coil inside diameter 3 [mm] Coil outside diameter 5 [mm] Coil length 10 [mm] Number of turns 100 Coil wire conductivity 6 10 7 [S/m] Wire cross-section 3.14 10-9 [m 2 ] The material used for the plate was Inconel with a conductivity of 1.02 10 6 S/m and relative permeability of 1. The Inconel plate had a thickness of 2.10mm and the lift-off (space between the probe and the specimen) was 0.65mm as in the real situation. The thickness of the plate can then be changed and the impedance of the coil calculated for each of the thicknesses. In the latter stage of the 3D modelling, the slots were introduced into the back surface of the plate and the impedance was calculated for the measured values of the ligament (this is a measurement of the remaining thickness above a slot, which can be considered representative of the ligament to a sub-surface defect). The slot dimensions were the same as the ones used experimentally and are illustrated in the following sections. 2.3 Meshing process The meshing process for the purpose of this paper had a very important role in the final results. While generally a bigger mesh means a less accurate result, a finer mesh will result in more computational time and memory usage, especially in 3D modelling (7). The other challenge for the meshing process in this project was the plate thickness variation (1) (8).This means the same mesh size would not result in the same number of mesh elements for different plate thicknesses. A convergence study on the mesh size was done to find the optimum number of mesh elements for a satisfactory range of results. 5
3. Experimental set-up For the experimental set-up in this paper, the coil dimensions used were as in Table 2. The coil was air-cored for the reasons explained earlier. The plate characteristics were as described in the modelling section. Figure 2 shows schematics of the plates. (b) Figure 2. (a) Inconel plate with EDM slots in the back surface of a plate with thickness t=2.10 mm; (b) Inconel plate with various thicknesses The eddy current instrument used was a Veritor eddy current flaw detector supplied by Ether NDE. The use of this particular equipment made it easier to measure the instrument output exactly because it is provided in a digital format. 6
3.1 Elements of a basic inspection system Figure 3 shows a block diagram of a basic eddy current system (9). It includes a single tone generator which energizes the test coil sensor. Phase, frequency and amplitude can be adjusted to the optimum values for the test pieces being used. When a change in the test piece occurs, the coil impedance experiences a change (10). The signal caused by the impedance variation is then filtered and amplified. The demodulator outputs are X-axis and Y-axis signals. Each component is derived from the real and imaginary parts of the impedance respectively. Figure 3. Elements of a basic inspection system 4. Experimental and simulation results The coil was balanced on a 2.10 mm thick Inconel plate and the lift-off signal was recorded for different plate thicknesses while the coil was excited using a 150kHz AC current. Figure 4 shows a typical screen output from the instrument during a scan from the EDM slot at the back surface of the plate. In Figure 4(a) the scan is performed at 50dB gain and Figure 4(b) shows the same scan with 65 db gain. It can be seen that the extra gain enables the operator to zoom in on the impedance plane variation in order to observe an effectively larger trace. 7
Figure 4.Typical screen output for scan of an EDM slot on the back surface (a) 50dB gain (b) 65dB gain The simulations were done with the coil held 0.65mm above the plate (lift-off) and the impedance of the coil was captured for different plate thicknesses at a frequency of 150kHz. results obtained from the experiments and the corresponding simulations are illustrated in Figure 5(a) and (b) respectively. 8
Figure 5. (a) Instrument readings (b) Simulation results Figure 5 indicates the differences between the results displayed on the screen of the instrument and the ones optained from the model. As explained in earlier sections, this difference is due to the functions of the control units available in the eddy current testing equipment, which modify the displayed phase and gain in order to enhance the visibility of small changes which, in practice, facilitate the interpretation of the results by the NDT operator. This meant that, although the simulation could be used to find out more information about the behaviour of a specific coil design and flaw type and impedance change from the flaw, the output shown on the instrument screen still cannot be directly predicted by the simulation software. When a typical test instrument is used, the coil impedance, which is a complex number containing both real and imaginary parts is transformed by translation (balancing), amplification (gain) and rotation (phase change) before it is displayed on the instrument. 9
In order to overcome this issue a series of basic mathematical trasformations were carried out, which are explained in more detail in the following section. This approach made it possible to calibrate the output of the instrument using a few calibration points, which then provide the means to model a new inspection situation and predict the corresponding output from the instrument. 5. Mathematical transformation For the purpose of converting the coil impedance to the equivalent instrument output, it was necessary to generate a number of calibration points from the instrument. Exactly the same calibration set-ups were also simulated by FEA. The two sets of results were mapped on to each other as closely as possible using the methods explained below: 1. The first step was to determine the centroid of each set of results and find the distance of each point from the centroid of each set. 2. The next stage was to find a least squares best fit for translation (balancing), rotation (phase) and stretching (gain) of the coil impedances from simulation to match the instrument X and Y coordinates. First, the slope and angle of the best fit straight line through each data set (an approximation to the overall trend) and the angle between these two lines was calculated. This angle is then used for rotation of the simulation results to match the instrument output. 3. A scale factor was then devised using the distances calculated in step 1 above, which minimised the least square distances between the simulated distances to the centroid of the (complex) coil impedance and the measured ones. The factors found here are then used to rotate and stretch the simulation results found by the software for a specific scenario to estimate the corresponding output from the instrument. 6. Results after transformation The results of the model output for plate thickness measurements, after applying the above algorithm, are shown in Figure 6. The results show good agreement for thickness measurements ranging from 0.4mm to 2.1mm. This provides some insight into the gain in db that would be needed to inspect a specific thickness. It also indicates the range of thicknesses that can be distinguished from each other depending on the specific sensitivity chosen for a test. The simulation for ligament measurement was then carried out. The results of these experiments and the transformed impedances obtained from the software are also shown in Figure 6. The results are again in good agreement for ligament measurement. The slight differences between the instrument reading and the transformed modelling results can be attributed to the limited ability of the software to accurately represent the true width of a very narrow slot (owing to the relative coarseness of the mesh) and the limited accuracy of the experimental positioning of the probe above the slot. Also, in general, the modelling of a given inspection scenario is somewhat idealised compared to the real situation, because the model does not account for the various sources of noise experienced by the instrument. 10
Y co-ordinate (Instrument Unit) X co-ordinate (Instrument unit) 10 0-10 -20-30 -40-50 -60-70 -80-90 -100 30 25 20 15 10 5 0 0.4 0.8 1.2 1.6 2 2.4 Plate thickness (mm) (a) Transformed real part of coil impedance (simulation ) Instrument Ligament (Instrument) Ligament (Transformed from model ) Transformed imaginary part of the coil impedance (Simulation) Instrument Ligament (Instrument) 0-5 0 0.4 0.8 1.2 1.6 2 2.4 Plate thickness (mm) Ligament (Transformed from model) (b) Figure 6 Instrument display (a) X co-ordinate (b) Y co-ordinate 7. Conclusions This paper describes a method that allows the output of a specific eddy current instrument to be estimated using a finite element model. This method provided reasonably accurate prediction of the output of the instrument for ligament measurement after the required transformations had been determined by calibration using thickness changes. Minor discrepancies between model and experiment may be due to the meshing process, noise in the experimental set-up of the instrument and small differences in the positioning of the coil. 11
This method will be useful in predicting the output of an instrument for an inspection situation where there are no previous measurements available. References 1. E. Narta and A. O. Ayhanb, Crack insertion, meshing and fracture analysis of structures using tetrahedral, European Journal of Mechanics A/Solids, no. 30, pp. 293-3036, 2011. 2. L. Wanga and T. Moriwaki, A Novel Meshing Algorithem for Dynamic Finite Element Analysis, Precision Engineering, no. 27, pp. 245-257, 2003. 3. A. Ammari, A. Buffa and J. C. Nedelec, A Justification of Eddy Current Model for the Maxwell Equations, SIAM J.APPL MATH, 2000, pp. 1805-1823. 4. J.Jin, The Finite Element Method in Electromagnetics, New York: John Wiley &Sons, 1993. 5. On the Use of the Magnetic Vector Potential in the Finite Element Analysis of Three-Dimensional, IEEE Transactions on Magnetics, vol. 25, no. 4, pp. 3145-3159, 1989. 6. O. C. Zienkiewicz, C. Emson and P. Bettess, A Novel Boundary Infinite Element, International Journal of Numerical and Methods in Engineering, pp. 393-404, 1983. 7. R.Verfurth, A Posteriori Error Estimation and Adaptive Mesh-Refinement, Journal of Computational and Applied Mathematics, no. 50, pp. 67-83, 1994. 8. G. Carey, A Perspective on Adaptive Modeling and Meshing (AM&M), Computer Methods in Applied Mechanics and Engineering, no. 195, pp. 214-235, 2006. 9. J. G. Martin, J. G. Gil and E. V. Sánchez, Non-Destructive Techniques Based on Eddy Current Testing, Sensors-Open access, vol. 11, pp. 2526-2565, 2011. 10. N. T. Hajian and J. Blitz, Prediction of Impedance Components for Eddy-Current Probe Coils for Standardization, NDT Internatinal, vol. 19, no. 5, pp. 333-339, 1986. 12