EDDY-CURRENT MODELING OF FERRITE-CORED PROBES
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1 EDDY-CURRENT MODELING OF FERRITE-CORED PROBES F. Buvat, G. Pichenot, D. Prémel 1 D. Lesselier, M. Lambert 2 H. Voillaume, J-P. Choffy 3 1 SYSSC/LCME, CEA Saclay, Bât 611, Gif-sur-Yvette, France 2 DRE-L2S (CNRS-SUPELEC-UPS), Plateau de Moulon, Gif-sur-Yvette, France 3 EADS CRC DCR/SE/IND, Suresnes, France ABSTRACT. The CEA has developed a fast code for the simulation of eddy-current probes. It is based on the Volume Integral (VIM) approach. Within the framework of an EADS partnership, this code has been extended to the simulation of different kinds of ferrite-cored probes with for example I, C or E shapes. The model gives the response of the probe to a parallelepiped 3D flaw included in a conductive slab. It has been benchmarked with experimental data for several configurations and an excellent agreement has been achieved. INTRODUCTION Eddy-current non destructive testing of conductive materials is of importance in many domains of industry and transportation. A simple typical configuration may consist of an axisymmetric ferrite-cored probe which is placed over a multi-layered conductive workpiece (conductivity varying typically from 0 between the metal parts to 60 MS/m inside them) and which is operating in the time-harmonic regime at frequency depending on the problem (typically between a few Hz to a few MHz). The aim of the ferrite core is to focus the magnetic fields into the workpiece, to increase the probe sensitivity to the defect. In this paper, a solution developed with a rigorous volume integral formulation based on the dyadic Green s functions is presented. The model obtained is linked to the NDT simulation platform CIVA [2] and used to compute in both reliable and quick fashion the normalized impedance diagram of a probe and the electromagnetic field distribution in the workpiece and in the magnetic core (if any) of the probe. THEORETICAL APPROACH A typical configuration of eddy-current testing is depicted in Fig.1. An axisymmetric probe is placed above a metal block (or a layered structure). It is made of a ferrite core, 463
2 typically cylindrical or E-shaped, excited by a coaxial coil (section area S) comprising n c turns and carrying electrical currents of density and total intensity I, their time dependence being implied. The ferrite-cored probe consists of two sources of current, an electrical one which is imposed by the user inside the windings of the coil, and a magnetic one which is induced in the core. The fields due to the electrical source and the corresponding impedance of the coil can be easily computed [1, 2]. On the other hand the fields and the impedance due to the core are more difficult to compute, even though the problem may be treated as a linear one (hysteresis is neglected) because of the weakness of the excitation imposed to the core. To carry out this calculation properly, closely following in that matter pioneering investigations [3], the core magnetization is introduced, the magnetic field within the core then satisfying the state equation where and are respectively the permeability of the vacuum and of the core, where is the primary field induced by currents and where is an appropriate magnetic-magnetic Green s dyad (field point and source point both located in air above the workpiece). The electrical conduction is neglected in the core but its permeability may be taken as complex-valued to take the magnetic losses into account. The stratification of the medium (if any, since it can also reduce to an half space) is treated by using generalized reflection and transmission coefficients at each interface in the Green s dyads [4]. A discrete solution of the state equation (1) is obtained by application of a Galerkin s variant of the Method of Moments [5], which involves a suitable discretization of the core into elementary cells in which the field quantities are assumed constant. Then, once is determined in each cell, the electric field in a given i th layer of the stratified workpiece easily follows by integration over the regions occupied by the sources (coil and core) from a discrete counterpart of (1) (2) where is an appropriate electric-electric Green s dyad (for which the field point lies in air whereas the observation point lies in the i th layer). The probe impedance in the presence of the workpiece follows from. (3) The axial symmetry of the sensor and workpiece is such that the primary electric field is azimuthal ( component only). So the magnetic field has two components (radial and longitudinal) only. As for the Green s dyads, they are reduced to scalar functions which are expressed in the form of semi-infinite integrals involving first-order Bessel functions. The next step is the introduction of the flaw inside the workpiece. Whereas the ferrite is modeled by considering a local permeability variation, the flaw is described by a local conductivity variation ( ) by comparison with the i th layer conductivity ( ). The presence of this conductivity variation involves a fictitious source of electrical 464
3 current. The whole electric field in the defect is obtained by adding the primary field to the electrical contribution due to the fictitious source (4) where the Green s dyad is an appropriate electric-electric dyad. The impedance variation of the probe due to the defect is then obtained by using the reciprocity theorem [6], and it reads as:. (5) QUANTITATIVE EVALUATION OF THE MODEL Three different configurations have been considered to quantitatively evaluate the model developed. The first case is a probe with a cylindrical ferrite-core above a half space. The second case is a probe with a E-shaped ferrite-core above a layered workpiece. The last case presents the response of a probe with a E-shaped ferrite core to a rectangular EDM notch in a slab. Probe with a Cylindrical Ferrite Core Above a Conductive Half-Space Let us consider a probe made of a coil (15.3 turns) with a cylindrical ferrite core (with a relative permeability ) as is shown in Fig.1. This probe is placed slightly above an aluminum half space ( MS/m). The core is discretized into 2 cells along the radial direction and 64 cells along the longitudinal direction. The magnitude of the magnetic field in the core, defined as and the one of the induced electric field inside the workpiece (I = 1 ma) for a 2 khz frequency are plotted in Fig.1. The probe performance can be assessed to some extent from the variation of its impedance above the workpiece,, with frequency. This is usually considered FIGURE 1. a- Fields in the core and the workpiece. b- Normalized impedance diagrams of the cylindrical ferrite-cored probe and the air-cored probe. 465
4 via a normalized impedance diagram, illustrating how R n = R/X$ and X n = X/X$ vary with frequency, XQ being the free-space reactance, R and X the resistance and the reactance of the probe above the workpiece. In Fig. 1-b the said diagrams obtained for the coil alone and for the ferrite-cored probe are compared to results given by a finite-element solver from Vector-Fields, operating frequencies varying from 1 khz to 1 MHz. In all cases, agreements between results obtained with our model (CIVA) and those obtained by the finite element method are better than 2 %, both for the real and imaginary parts. Another important result is the increase of the probe sensitivity when the core is introduced inside the probe. As a matter of fact, the field lines become much more confined and the coupling between the probe and the workpiece is better, which is providing an increase of Rn and X n. Finally, the electric field induced by the cylindrical ferrite-cored probe has been computed at a 2 khz frequency for two different depths (at the interface between air and metal and inside the workpiece at 2.5 mm). The magnitude of the E field (in V/m) as a function of the radius (in m) varying from 0 to 16 mm is displayed in Fig.2-a and in Fig.2- b. Again, agreements obtained in both cases are better than 1 %. E-shaped Ferrite-Cored Probe Above a Layered Workpiece The second configuration of study involves a coil (2450 turns) placed inside an E-shaped ferrite-cored (/^ = 2200) discretized into 12 cells along both radial and longitudinal directions and located above a four-layer medium as is depicted in Fig 3-b. The probe dimensions are given in Tab 1. Layers 1 and 3 are metal slabs with conductivity a = MS/m and thickness of 0.3 mm. Layer 2 is air and its thickness is 0.25 mm (the last layer, a half-space, is also air). I + I FEM I CIVA I \ ( E Radius (m) Radius (m) FIGURE 2. a- E field at z = 0 mm with a cylindrical ferrite-cored probe, b- E field at z = 2.5 mm with a cylindrical core probe : CIVA (in solid line) and FEM (+ results). 466
5 Coil FIGURE 3. Probe constituted by a coil placed inside an E-shaped core : a- slice view and dimensions of the probe, b- fields in the core and the workpiece. TABLE 1. Coil and E-shaped ferrite-cored dimensions. Coil (mm) External diameter: 7.60 Internal diameter: Height: 3.24 Number of turns 2450 Lift-off : 0.36 E shaped core (mm) H: 3.10 F: 7.48 E: A: B: 5.30 D: 3.60 D : 0 The impedance of the ferrite-cored probe above the four-layer medium and above a ten-layer medium 1 is computed and compared to experimental data. In all cases, agreements between results obtained with our model (CIVA) and those obtained experimentally are better than 2 %. 5 khz 5 khz 10 khz 10 khz 50 khz 20 khz FIGURE 4. Normalized impedance diagram of the probe computed with our model and determined experimentally. 1 The ten-layer medium is constituted with five metal slab (18.76 MS/m, 0.30 mm) separated from one another by one air-layer (0,25 mm), the last layer being air. 467
6 Coil (mm) External diameter: 7.60 Internal diameter: Height: 3.24 Number of turns 2450 Lift-off : 0.36 E-shaped core (mm) H: 3.10 F: 7.48 E: A: B: 5.30 D: 3.60 D : 0 FIGURE 5. a- Scanning area, b- Probe dimensions. Response of a Ferrite-Cored Probe to a Flaw The third configuration of study is of an E-shaped ferrite-cored probe ( = 2200) discretized into 12 cells in both radial and longitudinal directions and placed above an aluminum plate ( 14.1 MS/m, width 10 mm) containing an EDM notch. The notch has a length of 5 mm, a width of 0.1 mm and a depth of 0.8 mm. It is decomposed into 20 x 8 x 8 parallelepiped cells. The impedance of the probe is scanned over an area containing the notch as is depicted in Fig. 5. The impedance variation of the ferrite-cored probe when it is displaced along the length of the notch is computed at 2.5 khz 2 and is compared to experimental data in Fig. 6. For both real and imaginary parts, agreements between results obtained with our model (CIVA) and those obtained experimentally are better than 5 %. Same agreements are observed when considering two other frequencies 1 khz and 10 khz, respectively plotted in Fig. 7 and Fig. 8. FIGURE 6. Impedance variation of the probe along the length of the notch at 2.5 khz: a- real part, b- imaginary part. 2 This frequency corresponds to the maximum of the resistive part R n of the normalized impedance diagram of the ferrite-cored probe above the workpiece. 468
7 FIGURE 7. Impedance variation of the probe along the length of the notch at 1 khz: a- real part, b- imaginary part. FIGURE 8. Impedance variation of the probe along the length of the notch at 10 khz: a- real part, b- imaginary part. CONCLUSIONS The model developed and presented herein is based on the volume integral method, which has the advantage that only the ferrite core needs to be discretized and not the entire space in order to compute the impedance of the probe and the electric field in the workpiece. The model is reliable and fast (the numerical code is fully vectorized). Typically, three minutes are sufficient to compute all electromagnetic quantities of interest for the probe and the workpiece on a desktop computer PC with a CPU frequency of 2 GHz and a RAM of 528 Mo. The defect is introduced in the model using the same formalism (dyadic Green s functions) and the discretization is in this case reduced to the flaw. The global model handling a configuration of inspection with a ferrite-cored probe and a volumetric 3D flaw is validated by comparison with experimental data. The model is linked to the nondestructive testing platform CIVA and thus benefits of an NDT oriented graphical user interface and of numerous analysis tools. 469
8 REFERENCES 1. Dodd, C.V. and Deeds, W.E., J. Appl. Phys. 39, (1968). 2. Pichenot, G. and Sollier, T., Eddy-current modeling for non destructive testing, in Proc. 8 th European Conf. On Nondestructive Testing, Barcelona, Sabbagh, H.A., IEEE Trans. Magn. 23, (1987). 4. Chew, W.C., Waves and fields in inhomogeneous media, IEEE Press, Pictaway, Harrington, R.F., J. Electromagn. Waves Applicat. 1, (1987). 6. Bowler, J.R., Sabbagh, L.D. and Sabbagh, H.A., IEEE Trans. Magn. 26, (1990). 470
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