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     

Analysis of current distributions in a multi-laminated HTS tape conductor for solenoid coils

A multi-laminated HTS tape conductor has been recently developed for large coils. If the HTS tapes are simply laminated to form the conductor, the current distribution in the laminated tape conductor of the coil is imbalanced because of the differences among inductances of tapes. Transposition of the tapes in the conductor is effective for homogeneous current distribution, but the tape may be damaged due to the lateral bending. The solenoid coil has enough space to transpose the tapes at both ends. However, a proposed theory so far requires a restriction in the number of coil layers for homogeneous current distribution in the laminated tape conductor. It is very important to analyze current distributions in the multi-laminated tape conductor for the solenoid coil with arbitrary layers. In this paper, we apply the Maxwell integral equation to the region contoured by adjacent laminated tapes to analyze the current distributions of the tapes in an infinite solenoid coil, and demonstrate that the flux across the region is conserved as long as the tapes are not saturated, and finally induce the fundamental equations as functions of coil construction parameters, such as layer radii, laminated tape spaces, and winding pitches. We use the fundamental equations for 2-layer and 4-layer coils to verify the homogeneous current distribution of the laminated tape conductor for an arbitrary layer number. Since the flux between the tapes in the inner layer of a 2-layer coil is contributed from the outer layers, the tape space in the outer layer must be larger than that in the inner layer because of the balance between the two fluxes. Moreover, we have developed an analysis method for a finite solenoid coil.     

Benchmark problems for defect size and shape determination in eddy-current nondestructive evaluation

A set of four benchmark problems is presented for verification of theoretical calculations of defect size and shape in eddy-current nondestructive evaluation. The benchmark problems are based on careful measurements of the change in coil impedance as a function of frequency for a circular air-cored coil which is scanned along the axis of an electrodischarge machined slot in a thick aluminum alloy plate. Slots of (i) semi-elliptical, and (ii) double-peaked profiles are considered. Deviations from ideal coil behavior are identified and corrected so that the final impedance data and experimental parameters can be directly used to verify theoretical inversion algorithms.     

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     

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     

Measurements of current distributions in a multi-laminated HTS tape conductor for solenoid coils

A multi-laminated HTS tape conductor has recently been developed for large coils. If the HTS tapes are simply laminated to form the conductor, the current distribution in the laminated tape conductor of the coil is unbalanced because of the differences among all tape inductances. Transposition of the tape in the conductor is effective for homogeneous current distribution, but the tape may be damaged due to the lateral bending. In our previous paper, we proposed a new theory to analyze and control current distributions in the multi-laminated tape conductor for a solenoid coil with arbitrary layers. We applied the Maxwell integral equation to the region contoured by adjacent laminated tapes to analyze the current distributions of the tapes in the infinite solenoid coil, demonstrated that the flux across the region is conserved as long as the tapes are not saturated, and finally induced fundamental equations as functions of coil construction parameters, such as layer radius, laminated tape space, and winding pitch. In order to verify the theory, we designed two kinds of coils with homogeneous and inhomogeneous current distributions in the two-laminated tape conductor by adjusting the space between the tapes in the second layer, and fabricated them. In the case when the space between the tapes in the second layer is the same as that of the first layer, 0.31 mm in thickness, we measured the tape currents of 7:3 for the inner and outer tape of the first layer, respectively. We adjusted the space between the tapes of the second layer, 1.78 mm in thickness, while the space of the first layer remained unchanged, 0.31 mm in thickness. We obtained the homogeneous current distribution in the tape conductor. The experimental data were in good agreement with the theory.     

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.     

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.     

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.     

Analytical solution for impedance change due to flaws in eddy current testing

Green's function is used in order to derive the analytical solution for the change in impedance due to a presence of the flaws in a conductor. This solution is applied to a cylindrical flaw and a spherical flaw whose radii are much smaller than the radius of the test coil. For both cases, the change in impedance is obtained within Born's limit.     

Application of the hybrid finite element, finite difference technique to the solution of multistrand conductor problems

An algorithm for the application of a hybrid finite element, finite difference technique for the solution of linear conductor problems has been developed. This paper describes the implementation of the technique to single layer helically spiralling conductor strand geometries. Such conductor configurations occur in hollow conductors with magnetically neutral spacers. However, the technique can also be used as an approximation to the solution of expanded conductor or ACSR conductor problems if it is assumed that the core wires remain substantially magnetically neutral. The influences of conductor pitch and strand diameter or equivalent circuit parameters are illustrated by considering a number of different configurations. Theoretical samples which are physically similar to actual ACSR specimens are analyzed in order to infer the inductive role of the core strand.     

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     

Imaging the continuous conductivity profile within layered metal structures using inductance spectroscopy

This paper presents an inverse method for determining the conductivity distribution of a flat, layered conductor using a multifrequency electromagnetic sensor. Eddy-current sensors are used in a wide range of nondestructive testing applications. Single-frequency sensors are very common; however, the potential of an eddy-current sensor with spectroscopic techniques offers the ability to extract depth profiles and examine more fully the internal structure of the test piece. In this paper, 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 an inverse solution, a modified Newton-Raphson method was used to adjust the conductivity profile to fit a set of multifrequency inductances in a least-squared sense. The approximate Jacobian matrix (sensitivity matrix) was obtained by the perturbation method. Numerical results of the forward solution are provided for cases of step, continuous conductivity profiles. Good estimates for the conductivity profile were obtained. Experimental eddy-current tests are performed by taking the difference in inductance of the coil when placed next to a reference conductor and next to a layered conductor over the range 100 kHz - 1 MHz. Inverse results based on experimental and simulated data verified this method.     

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.     

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     

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.     

High-Frequency Giant Magnetoimpedance Measurement and Complex Permeability Behavior of Soft Magnetic Co-Based Ribbons

We present a method for extracting the magnetic parameters of a soft magnetic Co-based alloy (Co$_68.25$Fe$_4.5$Si$_12.25$B$_15$) in the high-frequency regime (100 MHz–2.5 GHz). The method uses the magnetic sample (as-cast ribbon) as the conductor of a terminated microstrip transmission line. It uses the complex propagation constant of the line, obtained from open/short circuit input impedance measurements, to extract the per-unit-length series impedance. It then uses a theoretical model, based on the distributed parameters and the skin effect, to extract the bulk transverse relative permeability. The model considers the geometrical configuration of the magnetic conductor (ribbon) when determining the series impedance as well as the complex transverse relative permeability$(mu_ tr= mu'_ tr- j mu'_ tr)$. We report on the effect of applying and varying an external dc magnetic field.     

Rapid inversion of eddy current data for conductivity and thickness of metal coatings

A feature-based method that determines the thickness and electrical conductivity of a coating on a metal plate from the change in the frequency-dependent impedance of an eddy-current probe coil is presented. Recently a least-squares solution of this problem was presented, which, however, requires approximately 20 CPU minutes on a DEC 5000 work station for the analysis of each set of measurements. We show that a feature-based approach can reduce the time to a few seconds on the same processor. We start by showing that a three-parameter scaling of the resistive component of the impedance change vs. frequency leads to a simple and nearly universal curve. Consequently these parameters provide a simple and compact way of expressing the data. Next, we show that the three scaling parameters can be used to construct a look-up table that determines the conductivity and thickness of the coating. Finally, we test the method using experimental data.     

Impedance Measurements Using Genetic Algorithms and Multiharmonic Signals

In this paper, a procedure to measure impedances using data-acquisition boards and genetic algorithms is developed. This approach to impedance measurements has the advantage of being low cost. The multiharmonic acquired waveforms are characterized using a genetic algorithm that finds the frequency of the signal, which, in turn, is used in a multiple linear least-squares (LS) waveform-fitting algorithm. The magnitude and phase of the unknown impedance can then be evaluated. A multiharmonic signal is used so that the frequency dependence of the impedance can be obtained from a single measurement. The measurement results are validated by measurements made with an impedance analyzer. The main advantage of the genetic algorithm over traditional search methods is its robustness to convergence problems.     

Modeling non-uniform frost growth on a fin-and-tube heat exchanger

A semi-empirical model that predicts non-uniform frost growth on heat exchangers is developed and experimentally validated. The model is based on a scaling approach that uses the average frost layer properties to predict growth in a quasi-steady state, heat and mass balance based segment-by-segment coil simulation. The air redistribution algorithm in the model improved frost thickness predictions by 20%-50% and coil capacity predictions by 42% compared to the same model without air redistribution. The model along with an empirical frost delay predicted the frost thickness for different inlet refrigerant temperatures, air relative humidities and air velocities under non-uniform frosting with a root mean square error of 3.7%, 9.8% and 5.2% respectively.