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     

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     

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     

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     

Measuring the temperature dependence of resistivity of high puritycopper using a solenoid coil (SRPM method)

The resistivity of high purity copper was measured by a method which estimates it by using the difference in the impedance between a circular multilayer solenoid coil with a circular rod-shaped copper sample and a similar coil without a copper sample (SRPM method). The residual resistivity ratio (RRR) of high purity copper measured at 100 Hz by the SRPM method has correlated well with the values measured by the DC four-probe method. It was confirmed that an accurate measurement of the resistivity to 10^-12 Ωm is possible by the SRPM method. Frequency dependence was confirmed to exist in high purity copper with very low resistivity. As the measuring frequency is raised, the decrease in skin depth seems to affect the resistivity     

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.     

Measurement of the magnetic susceptibility of a rod-shapedsuperconductor using a solenoid coil (SRPM method)

An effective method for the measurement of the magnetic susceptibility (χ) of a circular rod-shaped superconductor, in which χ is estimated by substituting the magnetic penetration depth (λ) into a formula after obtaining λ from the difference in the impedance between a solenoid coil with a sample and an identical one without a sample, is proposed. The temperature dependence of χ obtained from this method has correlated well with the values measured with a SQUID magnetometer     

A double coil method for simultaneously measuring the resistivity,permeability, and thickness of a moving metal sheet

A method to determine the resistivity, permeability and thickness of moving magnetic sheets is described. The parameters of a sample sheet inserted between two coils facing each other are determined by measuring the impedance of the two coils at two different frequencies. To compare the experimental data with theoretical values, the difference in the impedance of the coils between two cases is used: when the current passes through the coils in the same direction, and in opposite directions. The method was tested by measuring the resistivity, permeability and thickness of nickel and iron sheets, 1 mm thick, moving at velocities varying from 0 to 6 m/s. It was possible to accurately determine the resistivity, thickness, and permeability within 10%, 10%, and 20%, respectively, at frequencies of 400 and 800 Hz     

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.     

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 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.     

Thermal-Electromagnetic Analysis for 14 kA Fast Discharge of a Model Coil

A model coil for 40-T hybrid magnet superconducting outsert magnet has been constructed and tested at the High Magnetic Field Laboratory, Chinese Academy of Sciences. The model coil was wound with Nb₃Sn cable-in-conduit conductor (CICC) cabled in a 316LN jacket cooled with supercritical helium. The model coil alone can produce about 4 T maximum magnetic field with an operating current of 14 kA. The model coil, in combination with 7.57-T NbTi background coil, can produce 11.5 T central field at 14 kA. During the test campaigns, a fast discharge was triggered by a dump resistor of 3.6 mΩ to evaluate the thermal-electromagnetic behavior of the model coil. In order to avoid a quench of the background coil, no current was exerted on the background coil through a power supply during the fast discharge of the model coil. The test results show that the central magnetic field is not scaled proportionally to the current decay of the model coil. The circuit model gives excellent results compared with the measured ones for the central magnetic field evolution as a function of time in this paper. For the thermal-hydraulic behavior during the fast discharge, the maximum temperature at the inlet simulated by the 1-D Gandalf code gives excellent agreement results compared with the measured ones with the conductor coupling time constant of 63 ms.     

The Change in Impedance of a Single-Turn Coil Due to a Flaw in a Coaxial Conducting Cylinder

The problem of detection and location of a small flaw inside a conducting cylinder using an eddy current coil coaxial with the cylinder has been addressed. The electric field at an arbitrary axial and radial position inside the conductor has been obtained from a previous solution of the boundary value problem. An expression for the change in complex impedance due to a small flaw located within a conducting body has been derived and is shown to be a function of the electric field at the position of the flaw. For the case of a degenerate point flaw, this expression is further simplified by using just the value of the electric field at the position of the centroid of the flaw. The overall impedance is shown to be a function of the ratio of the radii of the loop and cylinder and of the conductivity of the material. The expression for the change in complex impedance has been factored into two terms, one dependent on the axial location of the flaw, and the other on the depth of the flaw. The axial location of the flaw is seen to affect only the magnitude and phase of the change in impedance; whereas the depth of the flaw is seen to affect both the magnitude and phase of the change in impedance. Plots of the complex change in impedance as a function of the axial location and depth of the flaw have been provided to illustrate its functional dependence on these parameters.     

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.     

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.     

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.     

Current displacement of a solid cylindrical conductor placed in a semiclosed slot, taking into account the leakage field in the gap

The field problem is solved assuming a constant magnetic field strength in the middle of the slot opening. Adapting the vector potential in gap and conductor, which has been expanded into Fourier series and described in different coordinate systems, one obtains a system of linear complex equations for the Fourier coefficients. The calculation of surface current density and surface field strength is checked by measuring the bar current and comparing it with a value deduced from the measurement of the voltage drop at the surface of the bar. This comparison becomes significant due to the fact that these quantities converge badly and determine the bar impedance by means of Poyntings's vector. Finally a comparison is made with the normal approach, which has high practical importance, and which takes the magnetic field strength on the surface of the conductor as a given boundary condition. This comparison shows the advantages of the analysis presented over the conventional approximation. At last a finite element (FE) solution for this problem is given.     

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     

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.     

Magnetic Field and Displacement Sensor Based on Giant Magneto-Impedance Effect

A two-core transducer assembly using a Fe_73.5Nb₃Cu₁Si_13.5B₉ ribbon to detect a change of magnetic field is proposed and tested for displacement (linear and angular) and current sensor. Two identical inductors, with the ribbon as core, are part of a two-series resonance network, and are in a high impedance state when excited by a small AC field of 1 MHz in the absence of a DC biasing field (H_dc). When the magnetic state of one inductor is altered by the biasing field, produced by a bar magnet or current carrying coil, an AC signal proportional to H_dc is generated by the transducer. The results for the sensitivity and linearity with displacement (linear and angular) of a magnet and with field from the current carrying coil are presented for two particular configurations of the transducer. High sensitivities of voltage response as much as 12 µV/µm and 3 mV/degree have been obtained for the transducer as a linear and angular displacement sensor, respectively, in the transverse configuration of exciting AC and biasing DC fields.