Electromagnetic eddy current sensors are commonly used to identify and quantify the surface notches of metals. However, the unintentional tilt of eddy current sensors affects results of size profiling, particularly for the depth profiling. In this paper, based on the eddy current thin-skin regime, a revised algorithm has been proposed for the analytical voltage or impedance of a tilted driver–pickup eddy current sensor scanning across a long ideal notch. Considering the resolution of the measurement, the bespoke driver–pickup, also termed as transmitter-receiver (T-R) sensor is designed with a small mean radius of 1 mm. Besides, the T-R sensor is connected to the electromagnetic instrument and controlled by a scanning stage with high spatial travel resolution , with a limit of 0.2 μm and selected as 0.25 mm. Experiments have been out on the voltage imaging of an aluminium sheet with 7 machined long notches of different depths using T-R sensor under different tilt angles. By fitting the measured voltage (both real and imaginary part) with proposed analytical algorithms, the depth profiling of notches is less affected by the tilt angle of sensors. From the results, the depth of notches can be retrieved within a deviation of 10 % for tilt angles up to 60 degrees.

The existence of the hole on a plate affects the calculation of eddy current problems. Consequently, the accuracy for the prediction of the material properties decreases if the effect of the hole is not taken into account. In this paper, a novel analytical method based on the modified integration range is proposed which can address the presence of the hole. Due to the presence of the hole, the conventional Dodd-Deeds analytical solution cannot be used to calculate the inductance change. Therefore, a revised upper integration limit is introduced to replace the original limit -- ∞ when using the co-axially air-core electromagnetic sensor. With the presence of the hole, the magnitude of the received signal reduces, and the peak frequency feature changes. The analytical method is validated by measured and numerical simulation results. It is found that the upper limit is related to the radius of the open hole. With the new technique, the thickness of sample plates with holes can be estimated based on the peak frequency feature.

This paper presents a new approach of eddy current methods for determining the size of the co-axial hole in the metallic circular disk. In recent decades, for the air-cored sensor probe, the impedance change due to the presence of an infinite metal plate can be calculated by the Dodd-Deeds model. However, in practical measurements, the sample cannot match with the condition required - ‘infinite’, thus the Dodd-Deeds model could not be applied to the disk with finite size and certainly not a co-axial hole in the center. In this paper, a dual-constraint analytical method is proposed. That is, the upper and lower limits of the integration are substituted with specific values instead of the original 0 and . Besides, it is found that, once the outer radius of the disk is fixed (i.e. the lower limit of integration is fixed), the upper limit reduces linearly as the size of the coaxial hole increases. Both the FEM simulation and experiments have been carried out to validate this method. The radius of the hole can be estimated based on the dual-constraint integration feature.

Electromagnetic sensing has been used for diverse applications of non-destructive testing, including the surface inspection, measurement of properties, object characterization. However, the measurement accuracy could be significantly influenced by the lift-off between sensors and samples. To address the issue caused by lift-offs, various strategies have been proposed for the permeability measurement of ferromagnetic steels, which mainly involves different sensor designs and signal features (e.g., the zero-crossing feature). In this paper, a single high-frequency scenario for the permeability retrieval is introduced. By combining the signal of two sensing pairs, the retrieval of magnetic permeability is less affected by the lift-off of sensors. Unlike the previous strategy on reducing the lift-off effect (directly taking the phase term out of the integration) using the Dodd-Deeds analytical method, the proposed method is based on a high-frequency linear feature of the phase term. Therefore, this method has the merit of high accuracy and fast processing for the permeability retrieval (a simplified version of Dodd-Deeds analytical formulas after the integration). Experimental measurement has been carried out on the impedance measurement of designed sensors interrogating ferromagnetic dual-phase steels. For sensor lift-offs of up to 10 mm, the error of the permeability retrieval is controlled within 4 % under the optimal frequency.

Eddy current testing can be used to interrogate steels but it is hampered by the lift-off distance of the sensor. Previously, the lift-off point of intersection (LOI) feature has been found for the pulsed eddy current (PEC) testing. In this paper, a lift-off invariant inductance (LII) feature is proposed for the multi-frequency eddy current (MEC) testing, which merely targets the ferromagnetic steels. That is, at a certain working frequency, the measured inductance signal is found nearly immune to the lift-off distance of the sensor. Such working frequency and inductance are termed as the lift-off invariant frequency (LIF) and LII. Through simulations and experimental measurements of different steels under the multi-frequency manner, the LII has been verified to be merely related to the sensor parameters and independent of different steels. By referring to the LIF of the test piece and using an iterative inverse solver, one of the steel properties (either the electrical conductivity or magnetic permeability) can be reconstructed with a high accuracy.

In many advanced industrial applications, the thickness is a critical index, especially for metallic coatings. However, the variance of lift-off spacing between sensors and test pieces affects the measured voltage or impedance, which leads to unreliable results from the sensor. Massive research works have been proposed to address the lift-off issue, but few of them applies to the thickness measurement of planar metallic films with finite-size circular (disk) geometry. Previously, a peak-frequency feature from the swept-frequency inductance was used to compensate the measurement error caused by lift-offs, which was based on the slow-changing rate of impedance phase term in the Dodd-Deeds formulas. However, the phase of measured impedance is nearly invariant merely on a limited range of sample thicknesses and working frequencies. Besides, the frequency sweeping is time-consuming, where a recalibration is needed for different sensor setups applied to the online real-time measurement. In this paper, a single-frequency algorithm has been proposed, which is embedded in the measurement instrument for the online real-time retrieval of thickness. Owing to the single-frequency measurement strategy, the proposed method does not need to recalibrate for different sensor setups. The thickness retrieval is based on a triple-coil sensor (with one transmitter and two receivers). The thickness of metallic disk foils is retrieved from the measured electrical resistance of two transmitter-receiver sensing pairs. Experiments on materials of different electrical conductivities (from direct current), thicknesses and planar sizes (radii) have been carried out to verify the proposed method. The error for the thickness retrieval of conductive disk foils is controlled within 5 % for lift-offs up to 5 mm.

Eddy current sensors can be used to test the characteristics and measure the parameters of the conductive samples. As the main obstacle of the multi-frequency eddy current sensor, the lift-off distance affects the effectiveness and accuracy of the measurement. In this paper, a material-independent algorithm has been proposed for the restoration of the lift-off distance when using the multi-frequency eddy current sensor, which is based on the approximation under the thin-skin effect. Experiment testing on the performance of the proposed method is presented. Results show that from the dual-frequency inductance, the lift-off distance could be restored with a maximum error of 0.24 mm for the distance up to 12 mm. Besides, the derived lift-off distance is used for the inversion of the magnetic permeability. Based on a lift-off insensitive inductance (LII) feature, the magnetic permeability of steels can be inversed in an iterative manner, with an error of less than 0.6 % for the lift-off distance up to 12 mm.

Lift-offs of the sensor could significantly affect the measurement signal and reconstruction of material properties when using the electromagnetic (inductive) eddy current sensor. Previously, various methods (including novel sensor designs, and features like zero-crossing frequency, lift-off point of intercept) have been used for eliminating the measurement error caused by the lift-off distance effect of the sensor. However, these approaches can only be applied for a small range of lift-off variations. In this paper, a linear relationship has been found between the sensor lift-off and ratio of dual-frequency eddy current signals, particularly under the high working dual frequencies. Based on this linear relationship, the lift-off variation can be reconstructed firstly with a small error of 2.5 % when its actual value is up to 10 mm (10.1 % for 20 mm). The reconstructed lift-off is used to further get the property of the material under a low single frequency. Experiments on different ferrous metals have been carried out for the testing of the reconstruction scheme. Since the inductance is more sensitive to the material property (and less sensitive to the lift-off) under low frequencies, the reconstruction error of the material property is much smaller than that of the lift-off, with 1.4 % under 12 mm (and 4.5 % under 20 mm).

Previously, various techniques have been proposed for reducing the lift-off effect on the thickness measurement of the non-magnetic films, including the peak-frequency feature and phase feature in the Dodd-Deed analytical formulation. To realise a real-time feedback response on the thickness monitoring, the phase term in the Dodd-Deeds formulation must be taken off the integration. Previous methods were based on the slow change rate of the phase term when compared to the rest of the term – the magnitude term. However, the change rate of the phase term is still considerable for a range of working frequencies. In this paper, a high-frequency feature has been found. That is, the ratio between the imaginary and real part of the phase term is proportional to the integral variable under high frequencies. Based on this proportion relationship, the phase term has been taken out; and a thickness algorithm has been proposed. By combing the measured impedance from the custom-built sensor (three coils), the thickness of the metallic film can be reconstructed. Experiments have been carried out for the verification of the proposed scenario. Results show that the thickness of the metal film can be reconstructed with a small error of less than 2 %, and immune to a reasonable range of lift-offs.

Electromagnetic sensors have been used for inspecting small surface defects of metals. Based on the eddy-current thin-skin regime, a revised algorithm is proposed for a triple-coil drive-pickup eddy-current sensor scanning over long surface crack slots (10 mm) with different rotary angles. The method is validated by the voltage measurement of the designed EC sensor scanning over a benchmark (ferromagnetic) steel with surface defects of different depths and rotary angles. With an additional sensing coil for the designed EC sensor, the defect angle (or orientation) can be measured without spatially and coaxially rotating the excitation coil. By referring to the voltage change (due to the defect) diagram (voltage sum versus voltage different) of two sensing pairs, the rotary angle of the surface crack is retrieved with a maximum residual deviation of 3.5 %.

The magnetic polarization tensor has a promising capability of determining the geometry and material properties of metallic samples. In this paper, a novel computation method is proposed to estimate the magnetic polarization tensors for the metallic samples using the boundary element method. In this method, the metallic sample is placed in a uniformly distributed magnetic field. Based on assumptions that the excitation frequency and/or the conductivity of the sample is very high, the metallic sample is regarded as a perfect electrical conductor (PEC). Therefore, the scattered field at a certain distance can be simulated. By utilising the boundary element method, the magnetic polarization tensor can be derived from the simulated scattered field. The theoretical calculation is presented and simulations and experiments have been carried out to validate the proposed method. The results from the simulation are matched with the analytical solution for the case of sphere samples. Moreover, there is a good agreement between the simulation results and the experimental results for the copper cylindrical samples.

Eddy current based approaches have been investigated for a wide range of inspection applications. Dodd-Deeds model and the truncated region eigenfunction expansion (TREE) method are widely applied in various occasions, mostly for the cases that the sample is relatively larger than the radius of the sensor coil. The TREE method converts the integral expressions to the summation of many terms in the truncated region. In a recent work, the impedance of the co-axial air-cored sensor due to a plate of finite radius was calculated by the modified Dodd-Deeds analytical approach proposed by authors. In this paper, combining the modified analytical solution and the TREE method, a new finite region eigenfunction expansion (FREE) method is proposed. This method involves modifying its initial summation point from the first zero of the Bessel function to a value related to the radius of the plate, therefore makes it suitable for plate with finite dimensions. Experiments and simulations have been carried out and compared for the verification of the proposed method. Further, the planar size measurements of the metallic circular plate can be achieved by utilising the measured peak frequency feature.

Previously, a conductivity invariance phenomena (CIP) has been discovered – at a certain lift-off, the inductance change of the sensor due to a test sample is immune to conductivity variations, i.e. the inductance – lift-off curve passes through a common point at a certain lift-off, termed as conductivity invariance lift-off. However, this conductivity invariance lift-off is fixed for a particular sensor setup, which is not convenient for various sample conditions. In this paper, we propose using two parameters in the coil design – the horizontal and vertical distances between the transmitter and the receiver to control the conductivity invariance lift-off. The relationship between these two parameters and the conductivity invariance lift-off is investigated by simulation and experiments and it has been found that there is an approximate linear relationship between these two parameters and the conductivity invariance lift-off. This is useful for applications where the measurements have restrictions on lift-off, e.g. uneven coating thickness which limits the range of the lift-off of probe during the measurements. Therefore, based on this relationship, it can be easier to adjust the configuration of the probe for a better inspection of the test samples.

A reliable and efficient rail track defect detection system is essential for maintaining rail track integrity and avoiding safety hazards and financial losses. Eddy current (EC) testing is a non-destructive technique that can be employed for this purpose. The trade-off between spatial resolution and lift-off should be carefully considered in practical applications to distinguish closely spaced cracks such as those caused by rolling contact fatigue (RCF). A multi-channel eddy current sensor array has been developed to detect defects on rails. Based on the sensor scanning data, defect reconstruction along the rails is achieved using an inverse algorithm that includes both direct and iterative approaches. In experimental evaluations, the EC system with the developed sensor is used to measure defects on a standard test piece of rail with a probe lift-off of 4-6 mm. The reconstruction results clearly reveal cracks at various depths and spacings on the test piece.