Analytical solutions to eddy-current testing problems for a layeredmedium with varying properties

A one-parameter family of analytical solutions is found for the case where a single-turn coil is located above a two-layer conducting medium. The conductivity σ and the relative magnetic permeability μ of the upper conducting layer are constants, while for the lower conducting half-space, these are of the form σ(z)=Az^b and μ(z)=Bz^a, where A and B are constants and a+b=-2 or a+b=0. The change in impedance is computed, and the results of numerical simulation are presented     

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.     

The quantitative relations between true and standard depth of penetration for air-cored probe coils in eddy current testing

The true depth of penetration of eddy currents generated in a conducting sample by an air-cored probe coil, besides depending on the electromagnetic wave frequency and the magnetic permeability and electrical conductivity of the sample, also depends strongly on the coil dimensions and the sample thickness. The standard depth of penetration widely used as a guide for eddy current inspection purposes is calculated for a plane electromagnetic wave incident perpendicularly on a conducting half-space and is thus a material/test parameter rather than a true measure of penetration. In this paper the quantitative relations between true depth of penetration and standard depth of penetration are presented for three configurations of eddy current probe and test material. First an air-cored coil above a conducting half-space is considered, then the same coil above a conducting sheet, and finally the true depth of penetration is calculated in a conducting half-space covered with cladding for different ratios of condictivity between cladding and base material.     

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.     

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.     

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.     

A model of eddy-current probes with ferrite cores

A model of a three-dimensional axisymmetric probe coil with a ferrite core in the presence of a conducting half-space (the workpiece) is developed. The half-space is accounted for by computing the appropriate Green's function by using Bessel transforms. Upon introducing equivalent Amperian currents within the core, we derive a volume integral equation, whose unknown is either the magnetic induction field, or induced magnetization, and whose kernel is the Green's function that was previously derived. The integral equation is transformed via the method of moments into a vector-matrix equation, which is then solved using a linear equation solver. This allows the computation of the magnetic induction field within the core, the driving-point impedance of the coil-core combination, and the induced eddy currents within the workpiece.     

Change in impedance of a single-turn coil due to a flaw in a conducting half space

The problem of detection and characterization of a flaw in a conducting half-space using an eddy-current coil oriented parallel to the interface is examined. An expression is derived for a first order approximation for the change in complex impedance due to a flaw located within the conducting medium. The overall impedance is a function of the radius and lift-off distance of the test coil and the conductivity of the material. An analytical expression is derived for the change in impedance as a function of the electric fields at the position of the flaw. It is found to be an integral over the volume of the flaw of the electric fields found with and without the flaw being present. The limiting case of a degenerate point flaw may be examined in greater detail by allowing the field in the presence of the flaw to be approximated by the unperturbed field. For flaws small enough that the field does not vary much over its volume, the field may be even further approximated by using just the value of the field at the position of the centroid of the flaw. Plots are shown to illustrate the behavior of the change in impedance as a function of the radial range of the flaw and the depth of the flaw centroid, using previously derived expressions for the fields for the unflawed case.     

Pulsed eddy-current response to a conducting half-space

Eddy-current nondestructive evaluation commonly carried out using single frequency time harmonic excitations, but a pulsed excitation offers a simple and effective alternative. The pulse signals have been calculated for a probe coil whose current rises and falls exponentially, approximating a square wave when the exponential time constant is small. Predictions of the induced electromotive force (EMF) across a coil above a half-space conductor and of the magnetic field on the coil axis have been compared with experiments. The comparison shows excellent agreement between theory and experiment     

Evaluation of Surface Impedance Models for Axisymmetric Eddy-Current Fields

We have investigated the range of validity of the perfect electric conductor and of the standard Rytov–Leontovich impedance boundary condition models for the analysis of axisymmetric eddy-current problems. Using these models, we derived approximate expressions for the magnetic vector potential, field quantities, Joule losses, and forces for conducting spheroids placed in external nonuniform magnetic fields produced by coaxial circular turns carrying currents varying sinusoidally with time. We compared our numerical results for the magnetic field intensity at the conductor surface, power losses, and forces (for both prolate and oblate spheroidally shaped conducting objects) with the results from analytical expressions obtained by applying the exact boundary conditions. While the simpler perfect conductor model can be employed only when the electromagnetic depth of penetration is much smaller than the smallest local radius of curvature, the results obtained by using the standard surface impedance model for conducting prolate and oblate spheroids of various axial ratios are in good agreement with the exact results for skin depths of about 1/5 of the semi-minor axis for electromagnetic forces and for skin depths less than 1/20 of the semi-minor axis for Joule losses.      

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     

Eddy currents induced in a finite length layered rod by a coaxial coil

We describe the calculation of eddy currents in a two-layer conducting rod of finite length excited by a coaxial circular coil carrying an alternating current. The calculation uses the truncated region eigenfunction expansion (TREE) method. By truncating the solution region to a finite length in the axial direction, we can express the magnetic vector potential as a series expansion of orthogonal eigenfunctions instead of as a Fourier integral. The restricted domain can be arbitrarily large to yield results that are as close to the infinite domain results as desired. Integral form solutions for an infinite rod are well known and relatively simple. For a finite length cylindrical conductor, however, additional boundary conditions must be satisfied at the ends. We do this by comparing series expansions term by term to match the solutions across the end of the cylinder. We derive closed-form expressions for the electromagnetic field in the presence of a finite two-layer rod. A special case of the solution is that for a conductive tube. We illustrate the end effect by calculating the coil impedance variation with respect to the axial location of the coil. The results are in very good agreement with those obtained by using a two-dimensional finite-element code.     

Analysis of diffracted fields with the extended theory of the boundary diffraction wave for impedance surfaces

Uniform diffracted fields from impedance surfaces are investigated by the extended theory of boundary diffraction wave (ETBDW). The new vector potential of the ETBDW is constructed by considering the pseudoimpedance boundary condition. The method is applied to the diffraction problem from an impedance half-plane. It is shown that the total fields from an impedance half-plane reduce to the case of a perfectly electric or magnetic conducting and opaque half-plane for special values of surface impedance. The total and diffracted fields are compared numerically with the exact solution for the impedance half-plane and modified theory of physical optics (MTPO) solution for an impedance wedge. The numerical results show that the field expressions are in very good agreement with the exact and MTPO solutions.     

Generalized electromagneto-thermoviscoelastic in case of 2-D thermal shock problem in a finite conducting medium with one relaxation time

Summary. This paper studies the problem of two-dimensional electromagneto-thermovisco-elasticity based on Lord-Shulman theory for a thermally and electrically conducting half-space solid whose surface is subjected to a thermal shock. There acts an initial magnetic field parallel to the plane boundary of the half-space. The medium deforms because of thermal shock and due to the application of the magnetic field, it results in induced magnetic and electric fields in the medium. The normal mode analysis is used to obtain the exact expressions for the considered variables. The distributions of the temperature, displacement, stress, induced magnetic and electric fields are represented graphically. Comparisons are made with the results predicted by both the coupled theory and with the theory of generalized thermo-viscoelasticity with one relaxation time.     

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     

Influence of the impedance to the surface currents induced on a spherical cap by a ring source

Explicit expressions for the high-frequency surface currents induced on a spherical cap having impedance property are obtained. It is shown that the induced electric current is always accompanied by a magnetic current as long as the impedance is different from zero. The results permit us to extend the application of the Physical Theory of Diffraction (PTD) to the cases of imperfectly conducting scatterers.     

The impedance of a single-turn coil near a conducting half space

The change in complex impedance between an ideal one-turn circular coil located above and parallel to a conducting half space with respect to a similar isolated coil has been calculated. From this result, a series expansion of the integrand allows the solution to be approximated by terms expressed as complete elliptic integrals. Results have been calculated for the change in impedance as a function of the liftoff distance and the conductivity of the half space for a coil of representative radius.     

Coil impedance due to a sphere of arbitrary radial conductivity andpermeability profiles

This paper presents analytical expressions for coil impedance due to a spherical workpiece consisting of concentric spherical shells. The expressions are used to simulate the nondestructive inspection of a sphere having arbitrary radial conductivity and magnetic permeability profiles by a circular coil of rectangular cross section. The simulation replaces continuous profiles with piecewise constant profiles. The paper compares the results to published experimental measurements and the results of other analytical solutions     

Modeling the surface condition of ferromagnetic metal by theswept-frequency eddy current method

Many ferromagnetic metals have, either by design or by happenstance, a very thin surface layer of reduced magnetic permeability. Such a layer might occur intrinsically or it might be induced by surface treatments such as case hardening, shot peening, or surface alloying. We studied the frequency-dependent impedance of a small air-cored coil of wire placed flat upon ferromagnetic metal plates. The impedance changes for nickel and iron could not be fitted to the half-space model for any values of conductivity and permeability. It is likely that the permeability varies continuously with depth in the near-surface region. Hence, it is possible that a multiple-layer model would yield a better representation of the data with a layer depth closer to that observed. In this paper, we illustrate that the multiple-layer model improves the agreement between the measured data and the approximate analytic solution. The results show that the permeability of pure nickel and pure iron decay exponentially from the base to surface     

Generalized magneto-thermoviscoelastic plane waves under the effect of rotation without energy dissipation

A new model of the equations of generalized thermoviscoelasticity for a thermally, isotropic and electrically conducting half-space solid whose surface is subjected to a thermal shock is given. The formulation is applied to the generalized thermoelasticity based on the Green and Naghdi (GN) theory under the effect of rotation, where there is an initial magnetic field parallel to the plane boundary of the half-space. The medium is deformed because of thermal shock and due to the application of the magnetic field, it results in induced magnetic and electric fields in the medium. The normal mode analysis is used to obtain the expressions for the variables considered. The distributions of temperature, displacement, stress, induced magnetic and electric fields are represented graphically. Comparisons are made with the results predicted by the types II and III in the presence and absence of rotation.