Evaluating the Permeability Distribution of a Layered Conductor by Inductance Spectroscopy

This paper presents an inverse method for determining the permeability distribution of a flat, layered conductor by using a multifrequency electromagnetic sensor. It discusses the use of spectroscopic techniques to extract depth profiles and examine more fully the internal structure of the test piece. The forward solution for a small right-cylindrical air-cored coil placed next to a layered conductor is based on the analytic solution provided by the transfer matrix approach. For the inverse solution, a modified Newton-Raphson method is used to adjust the permeability profile to fit a set of multifrequency inductances in a least-squared sense. The approximate Jacobian matrix (sensitivity matrix) is obtained by the perturbation method. The paper provides numerical results of the forward solution for cases of step and continuous permeability profiles. Inverse results based on experimental and simulated data verify the accuracy of this method, which yields good estimates for the permeability profile     

A Procedure for Measuring and Validating Samples of Electrical Conductivity

The possibility of using an eddy-current transducer to measure the electrical conductivity by an absolute (nonstandard) method, based on a solution of the inverse problem for the case of a coil with a current placed on the surface of a conducting nonmagnetic half-space is considered. A block diagram of the equipment, the construction of the eddy-current transducer, calibration curves and an analysis of the measurement errors are presented. The method is compared with the bridge method of measurement. __________ Translated from Izmeritel'naya Tekhnika, No. 9, pp. 59–65, September, 2005.     

Analytical solutions in eddy current testing of layered metals withcontinuous conductivity profiles

In this paper exact analytical expressions for the impedance of a cylindrical air-core coil above a layered metal structure whose conductivity varies continuously with depth are presented. Although the model is general, attention is focused on three conductivity profiles: the linear, the quadratic and the exponential. The derived expressions for the impedance change for each profile could provide a useful tool for the solution of the inverse problem: that of determining the conductivity from variable frequency measurements of the impedance. Furthermore, the obtained final formulas contain elegant mathematical functions and show a substantially higher computational efficiency with respect to existing methods     

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.     

Nondestructive Eddy Current Testing

When a coil is placed in proximity to a conductor, the magnetic field in the vicinity of the coil is altered by eddy currents in the conductor. The conductivity of a sample material may be determined by measuring the field. Closed-form solutions are presented for the magnetic field at the surface of a semi-infinite conductor for two cases: a semi-infinite coil, the end of which is in contact with the conductor, and a loop of wire on the surface of the conductor.     

Eddy current inversion for layered conductors

By making multifrequency eddy current measurements on a layered conductor, it is possible to acquire information on the depth dependence of the conductivity. We consider an inversion problem in which coil impedance data are used to determine either the layer thicknesses or layer conductivities. The algorithm is based on a well known forward model which gives the impedance of an air cored coil above a stratified conductor from a closed form expression. In the forward calculation, estimates of the unknown material parameters are used to get tentative predictions of the measurements. Differences between these predictions and measured impedances are expressed in terms of a global error that is minimized iteratively with the aid of a descent algorithm by varying the parameters of the structure. Examples of minimization searches for layer parameters are given.This article is dedicated to Professor Bertram A. Auld on the occasion of his 70th birthday and his retirement from Stanford University.     

Fast Methods for Quantitative Eddy-Current Tomography of Conductive Materials

In this paper, we address the imaging of the spatial distribution of the resistivity of conductive materials by using data from eddy-current nondestructive testing. Specifically, the data consists of measurements of the impedance matrix at several frequencies acquired using a coil array. The imaging method processes the second-order term (estimated from the measured data) of the power series expansion, with respect to frequency, of the impedance matrix. This term accounts for the resistive contribution to changes of the impedance matrix, due to the presence of anomalies in the conductor under test, occurring at relatively low frequencies. The operator mapping a given resistivity distribution inside the conductor into the second-order term satisfies a proper monotonicity property. The monotonicity makes it possible to apply a fast noniterative imaging method initially developed by the authors for elliptic problems such as electrical resistance tomography. Numerical examples show the main features of the proposed method, and demonstrate the possibility of real-time imaging.     

Mutual Impedance of Cylindrical Coils at an Arbitrary Position and Orientation Above a Planar Conductor

We derive a closed-form expression for the mutual impedance due to eddy-current induction for a pair of cylindrical air-core coils with arbitrary position and orientation above a planar conductor. By extending a recently devised model for individual coils with an arbitrary tilt with respect to the surface, we obtain a remarkably simple result. We validated our model with measurements on a conductive plate. The results should be useful for designing new probe configurations and for evaluating the signals in eddy-current inspections when driver-pickup coil configurations are utilized.     

An NDE Methodology to Predict Density in Green-State Powder Metallurgy Compacts

Extensive work at PMRC has established a clear correlation between green-state density and electric conductivity of P/M parts. By monitoring a static electric current flow through the pre-sintered P/M sample and recording the voltage response over its surface, sufficient information can be gathered to predict the density profile throughout the sample volume. In this paper, the formulation and implementation of a novel numerical forward and inverse formulation will be presented that is capable of relating DC voltage measurements to green-state density distributions. It will be shown that this methodology is applicable to both lubricated and lubricant-free compacts. We will present a general forward solution that enables the calculation of three-dimensional surface voltages for a given set of boundary conditions and a known conductivity/density distribution throughout the P/M sample. This formulation is particularly useful for the development of new sensors and measurement arrangements, since it permits the optimization of current injection patterns and voltage probe locations. Our novel inverse solution adjusts the conductivity/density profile so as to determine the conductivity distribution that matches most closely a given set of voltage data on the surface. Practical measurements with a range of green-state P/M samples will underscore the success and usefulness of this modeling approach.     

Three-dimensional eddy current analysis by the boundary elementmethod using vector potential

A boundary-element method using a magnetic vector potential for eddy-current analysis is described. For three-dimensional (3-D) problems, the tangential and normal components of the vector potential, tangential components of the magnetic flux density, and an electric scalar potential on conductor surfaces are chosen as unknown variables. When the approximation is introduced so that the conductivity of the conductor is very large in comparison with the conductivity of air, the number of unknowns can be reduced; also, for axisymmetric models the scalar potential can be eliminated from the unknown variables. The formulation of the boundary-element method using the vector potential, and computation results by the proposed method, are presented     

Radio frequency resonator measurements of aqueous solution concentration

In order to measure aqueous solution concentration, radio frequency resonator sensors based on sections of a long line are proposed: two-conductor, coaxial and screened with a U-shaped inner conductor. Specific sensor structural features and the contents of their radio frequency measuring devices are described. Results are provided for experimental confirmation of the efficiency of these sensors for measuring the concentration of various two-component liquids with losses. __________ Translated from Izmeritel’naya Tekhnika, No. 10, pp. 65–67, October 2007.     

Thermal Transport Properties of Layered Materials: Identification by a New Numerical Algorithm for Transient Measurements

Transient methods are widely used to determine thermal transport properties. In some situations they can be used for homogeneous media to measure several properties either simultaneously or separately. In this context an analytic model is available and a well-posed inverse problem of parameter identification has to be solved. The examination of composite media is more complicated. The algorithm proposed here allows simultaneous determination of the thermal conductivity and thermal diffusivity of layered dielectrics by transient measurements. It is based on a plane source that acts both as a resistive heater and temperature sensor. For the technique to be successful two essential aspects have to be considered: firstly, the mathematical modeling of the measured data (the forward problem) and secondly, the problem of ill-posedness of the inverse problem. For the proposed measurement configuration, a new fast data analysis algorithm based on an analytic solution for the forward problem is presented. In principle, a numerical solution such as an FEM solution of the heat conduction equation can be used instead of the analytical one, but the computational effort is much greater. The inverse problem is formulated as an output-least-squares problem, which leads to a transcendent algebraic system of equations. The method was successfully tested for different situations.Paper presented at the Fifteenth Symposium on Thermophysical Properties, June 22--27, 2003, Boulder, Colorado, U.S.A.     

Prediction and use of impedance matrices for eddy-current problems

Inclusion of both skin and proximity effects in the prediction of impedance matrices in eddy-current problems significantly complicates the prediction of the impedance matrix. In this paper, a technique employing a boundary element method is used to predict these impedance matrices using an additional constant vector potential which is added to the interior of every conductor. This constant vector potential is slightly altered for an axisymmetric problem and allows for the easy prediction of induced voltage in an eddy-current conductor. Perhaps the greatest contribution offered by this paper is in the interpretation of these matrices and in particular, with the negative components comprising the resistance matrix. The phasor diagrams, both for voltage and current as well as magnetic fields, are employed to aid in better understanding the information delivered within the impedance matrix. The explanations are directed specifically toward a three-coil axisymmetric problem. The technique is tested against the measured voltage in a three-phase current fed system     

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.     

A novel strain sensor based on graphene composite films with layered structure

The use of graphene for strain sensors has attracted enormous attention due to its prominent mechanical and electrical properties. In this paper, we report on the preparation and characterization of a novel type of strain sensor based on graphene composite films with layered configuration. Highly reliable and sensitive composite films strain sensors based on graphene were produced from solution processed graphene flakes by spray coating method. The layered strain sensor which could sustain a large tensile deformation (25% strain) demonstrated high sensitivity to mechanical strain with gauge factors of 6–35. And the sensitivity of this type of strain sensors can be tuned over a relatively wide range of values by adjusting the deposition parameters. What’s more, the layered composite films are more durable compared with the fragile pure graphene films. In addition the main mechanisms are investigated, resulting in theoretical models which predict very well the observed behavior.     

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.     

Performance of Magnetic Pulsed-Eddy-Current System Using High Dynamic and High Linearity Improved Giant MagnetoResistance Magnetometer

Eddy-current nondestructive testing is commonly carried out using single-frequency time-harmonic excitations. Pulsed excitation offers a simple and effective alternative to classical eddy-current techniques and is classically implemented with coil(s) (transmitter and/or receiver). We present a pulsed-eddy-current-system alternative using a high-performance giant magnetoresistance magnetometer is presented. An analytical solution of the magnetic flux density created by the transient eddy current is given. Comparison with experimental results shows that the approximations are accurate (1% or better). System performances are presented and discussed versus sensor characteristics and compared to classical ones (transmitter/receiver coil systems)     

A theoretical and computational model of eddy-current probesincorporating volume integral and conjugate gradient methods

A general three-dimensional computational model of ferrite-core eddy-current probes has been developed for research and design studies in nondestructive evaluation. The model is based on a volume integral approach for finding the magnetization of the ferrite core excited by an AC-current-carrying coil in the presence of a conducting workpiece. Using the moment method, the integral equation is approximated by a matrix equation and solved using conjugate gradient techniques. Illustrative results are presented showing the impedance characteristics and field distributions for practical eddy-current probe configurations     

High speed data collection system for magnetic field imaging by CTtechnique

A method of high-speed data collection used in imaging the magnetic-field distribution leaking from magnetic devices along any plane is proposed. The method uses the computed tomography (CT) technique. A high-speed data collection system developed for demonstrating the method is described. An image of the magnetic field is reconstructed from projection data provided by the voltage induced in the sensors of a line conductor and a rectangular coil moving in the plane of observation. Rotational scanning of the sensors is used for collecting the projections so that data-acquisition time can be reduced. To demonstrate the performance of the system, the reconstructed images of the magnetic-field distributions made by some permanent magnets are shown     

Method to Determine Tensile Stress Alterations in Prestressing Steel Strands by Means of an Eddy-Current Technique

The increased demand for reinforced concrete elements, for example, in bridge construction, requires long-term monitoring using multisensor techniques. Dynamic load inspection and maximum load detection is now an important requirement to avoid overstressing in light of past building failures in Germany, Austria, Poland, and Russia. Magnetoelastic force measurement of prestressed reinforcing concrete cables is also an investigative method for detecting steel failures (McMaster, 1986; Cullity, 1972; Fabo, 2002). An examination of coil impedance measurements made using eddy-current sensors placed on mechanically stressed steel specimens demonstrates a high dependency on the magnetoelastic effect. This stress measurement on prestressings or other reinforced concrete cables can be achieved by calibration of the material properties and stabilization of the magnetization state at the working point.