A benchmark problem for computation of ΔZ in eddy-current nondestructive evaluation (NDE)

Experimental results are presented for the change in coil impedance Z when a circular air-cored coil carrying an alternating current of fixed frequency is scanned across a rectangular slot in an aluminium alloy plate. It is proposed that these measurements be used as a benchmark test to verify theoretical calculations of probe response in eddy-current NDE.     

Eddy-Current Nondestructive Inspection with Thin Spiral Coils: Long Cracks in Steel

The impedance of a cylindrical coil and a planar circular spiral coil carrying an alternating current above (i) a defect-free conducting magnetic half-space and (ii) a conducting magnetic half-space containing an infinitely long slot with uniform depth and width is examined in detail. Closed-form expressions for the coil impedance in these cases are presented, based on the theories of Dodd and Deeds and Harfield and Bowler. The validity of these expressions is tested by measurements using steel plates over the frequency range 100 Hz-10 MHz. The theoretical predictions are in good agreement with experiment, with the best agreement for the smallest slot width. The results confirm that thin, flexible spiral coils offer some attractive features for eddy-current detection of cracks in metals, particularly in terms of sensitivity and potential for unobtrusive permanent attachment to the material being inspected. Approximate expressions for a spiral coil above a defect-free magnetic half-space are also given to allow easy calculation in limiting cases.     

Alternative approaches to the numerical calculation of impedance

In many practical applications of numerical analysis applied to low-frequency electromagnetic problems, the desired output is often in terms of coil impedance. The mesh variable calculated by the more commonly used numerical formulations is the magnetic vector potential from which quantities like field intensities, eddy currents and others are calculated. In nondestructive testing applications, the quantity of interest is often the impedance of a coil or an array of coils. This paper discusses the calculation of impedance as a post-processing computation and introduces a new method of calculation of impedances and inductances based on computation of energy in the finite element mesh. The results presented clearly show the advantage of using direct integration methods for 2D and axisymmetric geometries. The energy approach, while valid regardless of dimensionality, should be restricted to 3D applications. Multiple-coil configurations in 3D applications present a special problem in analysis. The total impedance or inductance can be easily calculated but not independent coil or differential impedances. A method for calculation of these quantities in 3D computations is also presented.     

Benchmark problems in rarefied gas dynamics

In order to identify the most efficient and reliable methods and solvers for modeling of rarefied gas flows, it is proposed to choose few benchmark problems to be solved by different methods. The main requirements to such problems, such as geometrical simplicity and small number of determining parameters, are formulated in the present work. Two benchmark problems are proposed. A comparison between numerical and experimental data of these problems available in the open literature is performed.     

Magnetic Field-Based Eddy-Current Modeling for Multilayered Specimens

Eddy-current inspection for nondestructive evaluation has traditionally been investigated in terms of coil impedance signals via theoretical and experimental methods. However, advanced eddy-current techniques use solid-state sensors such as Hall devices, giant magnetoresistive sensors, anisotropic magnetoresistive sensors, and superconducting quantum interference devices for magnetic field measurement to achieve better sensitivity and high temporal and spatial resolution in material evaluation and characterization. Here, we review the Dodd and Deeds integral model and use the truncated region eigenfunction expansion (TREE) method for computation of the magnetic field. This results in series expressions instead of integral ones. Thus, the computation is both simplified and speeded up so that it becomes convenient for solving one-dimensional eddy-current inverse problems. We compare the theoretical results from the analytical model with the results from a numerical simulation based on the finite-element method in terms of accuracy and computation time.     

Constrained optimization using a multipoint type chaotic Lagrangian method with a coupling structure

This article proposes a new constrained optimization method using a multipoint type chaotic Lagrangian method that utilizes chaotic search trajectories generated by Lagrangian gradient dynamics with a coupling structure. In the proposed method, multiple search points autonomously implement global search using the chaotic search trajectory generated by the coupled Lagrangian gradient dynamics. These points are advected to elite points (which are chosen by considering their objective function values and their feasibility) by the coupling in order to explore promising regions intensively. In this way, the proposed method successfully provides diversification and intensification for constrained optimization problems. The effectiveness of the proposed method is confirmed through application to various types of benchmark problem, including the coil spring design problem, the benchmark problems used in the special session on constrained real parameter optimization in CEC2006, and a high-dimensional and multi-peaked constrained optimization problem.     

Impedance calculation of a single-layer air-core coil by means of resonant modal theory

A method for calculating the impedance of a single-layer air-core coil is developed. The coil is expressed by a pair of multiphase distributed-parameter lines. The nodal voltages within the coil are transformed into a modal domain by applying an eigen theory. The transformation diagonalises the nodal equation expressed by Y-parameters of the distributed-parameter lines. The method permits composition of an equivalent circuit of the air-core coil. The accuracy of the model is easily tuned by adjusting the number of eigenvalues taken into account. The choice of the eigenvalues is carried out by the voltage distribution along the wire of the coil. The dominant mode is obtained by an assumption that the voltage distribution is uniform. This mode gives the lowest antiresonant frequency of the coil. The other distribution patterns are assumed to be sinusoidal. The second dominant mode corresponds to the lowest space frequency among the distribution patterns. In general, there is no relation between the order of the calculated eigenvectors and the voltage distribution. Its order is sorted according to the voltage distribution using an initial eigenvector matrix. The accuracy of the impedance increases with the number of eigenvalues, which are taken into account according to the sorted order. A simplified lumped-parameter equivalent circuit of the coil is derived for the major modes. The simplified model is able to represent not only antiresonances but also resonances of an air-core coil. The accuracy of the proposed model is confirmed by comparisons with measured and theoretical results.     

An investigation of horizontal axis coils for eddy current inspection

The possible use of horizontal axis eddy current coils for the inspection of fast breeder reactor primary vessels is being investigated. An approach which considers both theoretical modelling work and experimental work has been adopted. Owing to the lack of analytical theory for the horozontal coil case, the initial work has concentrated on determining the impedance change of a horizontal axis air-cored coil when it is brought close to a conducting half-space. An analytical solution (due to Burke) has been considered, along with a newly developed approximate model, for a range of different coils above various conducting media. The approximate model assumes a uniform H field at the material surface and has been extended to consider layered half-spaces such as stainless steel over liquid sodium. The trends exhibited by the impedance change values obtained from both theories compared well with the experimental data.     

The study of SPRM method for a short-length rod-shaped sample

The SRPM (simultaneous resistivity and permeability measurement) method, which simultaneously estimates the electrical and magnetic properties of a rod-shaped sample conductor, has been studied. In the SRPM method, the theoretical and measured values of the difference in the complex impedance of a solenoid coil with and without the sample are used. To estimate the resistivity of a short-length sample, the difference in the complex impedance was calculated by the method of weighted residual (MWR), as the difference calculated by the former SRPM method was valid only for a longer sample. The resistivity of copper samples of various lengths was estimated by the SRPM method, and their ratios to those measured by the DC four-probe method were obtained. The ratio was less than 1.3% when the ratio of the length of the sample conductor to the length of the solenoid coil was larger than 1 and less than 3.1% when the length ratio was less than 1     

A normalization scheme for comparing eddy current differential probe signals using a finite element formulation

A first-order finite element formulation is used to model an eddy current differential bobbin coil probe scanning a tube with axisymmetric flaws. A multifrequency signal normalization scheme is developed to allow direct comparison between experimental measurements of the differential bobbin coil probe signal and finite element calculations of the probe coil impedance. Results demonstrate that both magnitude and phase of the differential bobbin coil impedance are useful in characterizing flaws in tubing for multifrequency scans.     

Eddy-current interaction of a long coil with a slot in a conductive plate

We describe a truncated-domain method for calculating eddy currents in a plate with a long flaw. The plate is modeled as a conductive half-space and the flaw is a long slot with a rectangular cross section. A long two-dimensional (2-D) coil carrying an alternating current is aligned parallel to the slot. The coil impedance variation with frequency is determined for an arbitrary coil location. The electromagnetic field due to a long coil above a conductive half-space can be expressed as integrals of trigonometric functions. For a half-space with a long slot, however, additional boundary conditions must be satisfied at the slot walls. The truncated-domain method makes this possible by recasting the problem in a finite domain; as a result, the Fourier integral is replaced by a series. The domain can be made arbitrarily large, thereby yielding results that are numerically as close to the infinite domain solution as desired. We have used the truncated domain approach to study both eddy-current flaw interactions and edge effects in the limiting case of a very wide and deep slot. We confirmed the theoretical predictions by comparing them with results of a 2-D finite element calculation and of experiments.     

Analytical model for tilted coils in eddy-current nondestructive inspection

The electromagnetic field and impedance of a cylindrical eddy-current probe coil are calculated analytically for arbitrary coil orientation above a conductive half-space. The remarkably simple closed-form expressions are provided as a function of coil tilt angle. The effect of tilt on the impedance change produced by a long crack is also investigated by combining the analytical model with an existing thin-skin theory for surface crack inspections. Results for both cracked and uncracked conductors are expected to be useful for evaluation of movement-generated noise in eddy-current inspections.     

Series solution of an eddy-current problem for a sphere withvarying conductivity and permeability profiles

Two-parameter families of analytical solutions are found for a single-turn coil symmetrically located above a two-layer sphere. Several cases are considered where the conductivity and relative magnetic permeability of the outer spherical layer are functions of the distance ρ from the sphere's center. The change in impedance of the coil is obtained in terms of a series containing Bessel functions. Computational impedance results are presented for different values of the parameters of the problem and the exact conductivity and permeability profiles are given diagrammatically for comparison with the impedance results     

A method for simultaneously measuring resistivity and permeabilitywith a multilayer solenoid coil

A formula is given for a more accurate estimate of either the resistivity and permeability of a cylindrical magnetic material or the resistivity and penetration depth of a cylindrical superconductor, using the difference in the complex impedance between a circular multilayer solenoid coil having a conductor and a similar coil without a conductor. In comparison with the conventional method which uses a single-layer solenoid coil, it is shown experimentally that a multilayer solenoid coil is more effective for a short sample conductor because a larger difference in the complex impedance is obtained without increasing the width of the solenoid coil. The results of measurements of magnetic and superconductive materials are included     

Reconstructing electrical conductivity profiles from variable-frequency eddy current measurements

A method for reconstructing radially varying conductivity profiles in cylindrical conductors is described. Solenoidal driving and sensing coils surround the cylindrical sample and an AC magnetic field applied by the driving solenoid induces axisymmetric eddy currents in the sample. It is shown how a radially varying conductivity profile can be recovered from measurements of the complex impedance recorded as a function of frequency, where impedance here is defined as the ratio of the induced electromotive force (EMF) in the sensing coil to the current in the driving coil. An iterative nonlinear least-squares algorithm is employed to reconstruct the profiles. Demonstrations of the reconstruction method are presented based on both simulated and experimentally recorded impedance data.     

Determination of Thickness of Silver Coatings on Brass by Measuring the Impedance of a Thin Elliptic Coil

To measure the impedance of a thin elliptically shaped coil, in presence of a flat plate with a coat of metal, can be an instrument for determination of the cladding thickness. An electromagnetic field from the coil is then forced to the object, producing eddy currents inside the object. These are influenced by the characteristics of the object and the coil and give rise to an impedance change, which can be detected and correlated to the thickness of the coating. An electromagnetic model accounting for the impedance of the elliptic coil with different values on the numerical eccentricity and the coating thickness is described. The model is based on a dyadic Green function formulation of the problem from which the electric field and hence the impedance is evaluated by utilizing the method of scattering super position. Numerical calculations based on the model and experimental measurements have been taken. An example shows how the model can be used to model a brass surface with a coat of silver to find expected impedance as function of the coating thickness     

A new optimization algorithm for solving complex constrained design optimization problems

This article presents the performance of a very recently proposed Jaya algorithm on a class of constrained design optimization problems. The distinct feature of this algorithm is that it does not have any algorithm-specific control parameters and hence the burden of tuning the control parameters is minimized. The performance of the proposed Jaya algorithm is tested on 21 benchmark problems related to constrained design optimization. In addition to the 21 benchmark problems, the performance of the algorithm is investigated on four constrained mechanical design problems, i.e. robot gripper, multiple disc clutch brake, hydrostatic thrust bearing and rolling element bearing. The computational results reveal that the Jaya algorithm is superior to or competitive with other optimization algorithms for the problems considered.     

Eddy-Current Coil Interaction With A Perfectly Conducting Wedge of Arbitrary Angle

In this article, a closed-form expression for the impedance of a tangential eddy-current coil in the presence of an infinite conducting wedge of arbitrary angle is derived. The truncated eigenfunction expansion (TREE) solution given here is valid in the quasi-static frequency regime. The theory was validated via comparison to an independent analytical expression for the impedance change of a horizontal coil over a conducting half-space due to Burke. We present results for three geometries: a conducting quarter-space, a conducting wedge of angle 225 degrees, and a semi-infinite conducting sheet. Our theory predicts a measurable change in the tangent coil reactance in the presence of all three geometries.     

Simultaneous measurement of electric and magnetic properties of aspherical sample

A method using a solenoid coil for simultaneously estimating the electric and magnetic properties of a spherical conductor was studied. These properties are estimated by finding the difference in the complex impedance of the coil with and without a sample to find out the measuring value that best coincides with the theoretical value. A new formula applicable to a nonmagnetic, a magnetic or a superconductive spherical sample was derived. The conductivities a and permeabilities μ measured by this method and by the conventional methods were compared using various samples. The deviations were no larger than 3% for a nonmagnetic samples, and 1.5% for μ of magnetic samples     

Impedance Resonance: A Novel Technique for Signal Acquisition From Interdigitated Electrodes (IDE) in Sensor Applications

A simple and useful modification to the common method of acquiring the impedance spectrum of modified interdigitated electrodes consists of including an appropriately sized induction coil connected in parallel with the interdigitated electrode (IDE). The resulting impedance behavior of the IDE/coil combination transforms the customary Bode plot of total impedance versus frequency data from a broad, complex curve to a simple resonance peak which can serve as the basis for a sensor output signal. The exact frequency and amplitude of the impedance resonance peak are a function of both the fixed induction characteristics of the coil and the experiment-dependant capacitance of the modified IDE. Quantifiable changes in the impedance resonance peak have been correlated with experimental parameters such as the electrical characteristics of polymer coatings on the IDE array leads, exposure of the IDE to aqueous solutions of varying composition, and molecular mobility factors in curable polymer resins.