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     

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     

Impedance of a single-turn coil due to a double-layered sphere withvarying properties

This paper presents an analytical solution for the change in impedance in a coil due to the presence of a conducting double-layered sphere symmetrically situated with respect to the coil. In the case where the relative magnetic permeability, μ(r)=r^α, of the outer sphere is a function of the distance r from the center of the sphere, where a is an arbitrary real number, the solution is expressed in the form of a series containing Bessel functions and can be used to compute the change of impedance. Numerical results for the case α=-2 are presented     

A novel planar mesh-type microelectromagnetic sensor. Part II. Estimation of system properties

The use of planar-type sensors for the estimation of system properties has gained considerable importance in recent times because of its noncontact and nondestructive nature. The impedance of a coil in proximity of any conducting/nonconducting, magnetic/nonmagnetic surface is a complex function of many parameters, such as conductivity, permeability, and permittivity of near-surface materials, liftoff and coil pitch of the coil, etc. The transfer impedance (i.e., the ratio between the sensing voltage and the exciting current) of the planar-type microelectromagnetic sensors consisting of exciting and sensing coils is used for the estimation of the near-surface system properties. Two methods have been discussed for the postprocessing of output parameters from the measured impedance data. Based on the estimation of near-surface properties, it is possible to detect the existence of defects, to predict the degradation of material, fatigue, etc.     

The Electromagnetic Basis for Nondestructive Testing of Cylindrical Conductors

Using an idealized model, we deduce the impedance per unit length of long solenoid of many turns that contains a cylindrical sample. The sample with a specified conductivity and magnetic permeability need not be centrally located within the solenoid provided all transverse dimensions are small compared with the free-space wavelength. The derivation is relatively straightforward and it provides a justification for earlier use of the impedance formula. The dual problem , where the solenoid is replaced by a toroidal coil is also considered. It is shown that both excitation methods have merit in nondestructive testing procedures.     

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     

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     

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.     

Simultaneous measurement of the resistivity and permeability of afilm sample with a double coil

A method is proposed for a simultaneous measurement of the resistivity and permeability of a film inserted between two coils facing each other. Connecting in series, the total impedance of the coils was measured in two ways, i.e., 1) the current passes through the coils in the same direction, and 2) the current passes through each coil in opposite directions to each other. The resistivity and permeability of the film were simultaneously obtained from the difference in the impedance for the two cases. It was theoretically found that the optimum frequency for high accuracy in the measurement was proportional to the resistivity and inversely proportional to a thickness of a sample film. The results of the simultaneous measurements of the resistivity and permeability of nickel films of a thickness ranging from 0.01-0.08 mm at a frequency range of 1-100 kHz are shown in this paper. The measured values of the resistivity and permeability with this method agreed with the values obtained by conventional methods, i.e., the dc four-probe method for the resistivity measurement and the toroidal coil method for the permeability measurement     

The Reflected Impedance of a Circular Coil in the Proximity of a Semi-Infinite Medium

Most analyses on a circular coil when used in the eddy current method for nondestructive testing are empirical. Theories based on simple models are often inadequate to account for some experimental observations when the spacing between the coil and the material became small. In the present paper this problem is formulated as a boundary value problem. Wave equations of the magnetic vector potential are solved. The change in the coil impedance, when placed above a semi-infinite medium, is obtained by means of the induced voltage method, which is shown to depend only on the ? component of the magnetic vector potential. This change in impedance is found to be dependent on a number of factors: the shape and size of the coil; the spacing between the coil and the metal; the thickness, conductivity, and composition of the material, etc. Numerical computations are discussed for a few selected materials in connection with experimental results obtained elsewhere. The comparison made lent support to the present analysis. Extension of this method to the case of a stratified media is included.     

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.     

Impedance of single-turn coil surrounding a conducting cylinder with a flaw

The impedance of a single-turn coil which surrounds a conducting cylinder with a flaw is calculated employing Green's function technique. The Born approximation is used in order to know how the impedance change due to the presence of a flaw depends on the conductivity and size of a flaw.     

A field-theoretical approach to magnetic induction heating of thin circular plates

A field-theoretical approach is used to analyze the subject of magnetic induction heating of thin circular plates by planar coils. Closed-form solutions for the electric and magnetic fields are found to the basic field problem of a single circular loop carrying current at a frequency ω in the presence of a plate characterized by a permeability μ and conductivity σ. By using these fields, expressions are derived for the complex Poynting vector at the surface of the plate, and for the induced EMF in the coil. The theory is extended to include multiturn coils and a field-dependent permeability, and a specific multiturn coil and plate combination is chosen as an example. The complex amplitude of the magnetic field and the Poynting vector are calculated along the surface of the plate using iterative methods to assure self-consistency with the field dependent permeability of the plate. By using Fourier transform techniques, the transient coil current and coil voltage waveforms are calculated under the experimental conditions used to take data on the sample coil and plate. The absorbed power is calculated from these waveforms and is found to be within 10 percent of the measured power absorption for all levels of operation from 50 to 2000 W. The calculated coil current waveform is compared with the measured waveform and is found to be in very good agreement in both shape and period.     

Calculation of the difference in impedance for a solenoid coil withand without a sample conductor

Either the resistivity and permeability of a magnetic material or the resistivity and magnetic penetration depth of a superconductor can be simultaneously estimated from the difference in the complex impedance between a circular solenoid coil coaxially surrounding a cylindrical conductor and an identical coil without a sample conductor. A method for calculating the difference in the complex impedance at a high frequency including the displacement current to an accuracy of 0.1% is reported. Comparison of the values calculated by this method and the values obtained by a conventional method which does not include the term of displacement current is also included     

Eddy current measurement of the electrical conductivity and porosity of metal foams

This paper presents experimental results characterizing the electrical properties of metallic foams, a relatively new class of material, using nondestructive eddy current sensing techniques. The fundamentals of eddy current sensing, which is based on electromagnetic induction, are described, and the effects on coil impedance change of the representative types of coil sensors are analyzed. It has been found that the phase-frequency response of the normalized eddy current signal of the sensor is relatively immune to coil-to-sample spacing and fill-factor variations, from which key results such as the equivalent conductivity and the porosity of the foams are presented. The paper demonstrates the broad applicability of this technique in characterizing and further recognizing the properties of a variety of sample shapes used.     

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.     

Electromagnetic Nondestructive Testing of Cylindrically Layered Conductors

We extend a previous analysis for the series impedance of a long solenoid to allow for the cylindrical layering of the encircled conductor. The results are discussed in the context of nondestructive testing of steel ropes that may have external or internal corrosion. It is shown that even an internal layer of reduced conductivity and permeability will be detectable if the frequency is sufficiently low to permit penetration of the primary field.     

Calculation of self- and mutual impedances in planar sandwichinductors

High-frequency planar magnetic components, employing thin film and thick film technology, have become important components in applications, such as filters and switching converters, due to their ease of manufacture and reliability. In a previous paper, the authors established a frequency dependent impedance formula for planar coils on a magnetic substrate that is infinitely thick. In this paper, two new impedance models are described: the first is for planar coils on a magnetic substrate of finite thickness, and the second represents a planar coil sandwiched between two substrates. The models include the electrical conductivity of the magnetic material so that the effects of eddy currents, particularly at high frequencies, are taken into account. The eddy currents reduce the inductance and increase the losses associated with the device. The new impedance formulas are derived from Maxwell's equations. Simulations were carried out on a typical device, using finite element analysis, and the results validate the new formulas. This paper establishes the frequency limitations of lossy magnetic substrates     

Analytical equivalent impedance for a planar circular induction heating system

A set of analytical expressions is derived for the equivalent impedance in a planar circular induction heating system, used for example in home appliances. The induction system consists of an n-turn planar winding loaded by a conductive material. Expressions that describe the frequency dependence of impedance are provided. The influence of load conductivity, load permeability, and geometrical dimensions is also considered. Validation of the proposed model is carried out through experimental measurements.