Towards an Universal Method for Predicting EddyCurrent Sensor Characteristics in the Railway Industry
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Abstract
Eddy current (EC) sensors placed in the vicinity of a rail track and sensitive to passing train components are safetycritical components commonly used in the railway sector. From the engineering point of view they form a system with numerous variables including their geometry, the liftoff, magnetic properties of the applied materials, the operating frequency and electrical characteristics of the builtin circuits. Although some simple configurations of the EC devices have been studied experimentally, numerically and analytically, there have so far been no universal algorithm allowing for predicting, understanding and optimizing output signals for arbitrarily device/track/wheel characteristics. A significant step towards such an universal algorithm is presented in this work, combining a linear 3D finite element modelling and a set of analytical formulas derived directly from the constants of the resonance circuit. Functions correlating the magnetic induction space/time characteristics with the circuit outputs are proposed and validated against experiment in three stages: (a) by determination of the signal from a flat thin FeSi plate (i.e. in fully controlled laboratory conditions), (b) by laboratory measurements of a highly degraded rail with uncertain material properties, (c) by tests of a commercial EC wheel detector, with reference to averaged service data. The proposed methodology can be extended to any NDT field where EC devices are applied.
Keywords
Eddycurrent testing Electromagnetic axle counters Finite element modelling Resonant circuits1 Introduction
The eddycurrent (EC) based sensors are widely used in Nondestructive Testing in the railway industry. EC defect detectors, as well as electromagnetic axle counters (a.k.a. “wheel detectors” or “train wheel sensors”) are safetycritical devices which require a thorough understanding in order to correctly interpret noisy signals and avoid both falsepositive and falsenegative indications. However, the complexity of the problem is such, that in sensorproducing companies known to the authors the empirical trialanderror approach tends to prevail over comprehensive theoretical insight. Even if undisturbed by noise sources, the EC sensor output depends on numerous factors, such as the coil geometry, circuit configuration, operating frequency, shape of the measured object, its electromagnetic properties (possibly nonhomogeneous and nonlinear), and presence of metallic auxiliary parts within the device. The surface condition and material properties have also an influence on the EC measurement result.
The EC NDT devices with resonant RLC circuits have been devoted numerous publications, including but not limited to the railway context. Some representative examples are mentioned below. Several authors have examined the laboratory or in situ the properties of the excitation coil(s) and the entire device, determining the sensitivity to the material defect, liftoff influence, or proposing better signal analysis schemes (e.g. [1, 2, 3, 4, 5]). The purely experimental and statistical approach does not, however, provide insight into underlying phenomena. In particular, it does not determine the relationship between the magnetic field distribution and the output signal dynamics.
Another approach consists in numerical [usually finite element analysis (FEA)] reconstruction of the amplitudes and phases of magnetic fields and ECs within the target object and in the surrounding space. The modelling is usually accompanied by experimental validation and aims at guiding the further development of the EC NDT methodology. A representative modelling and experimental work by Rocha et al. [6] focuses on performance of magnetic probes tested in velocity EC testing. In another research [7], Zhu et al. model a rail with a nonhomogeneous magnetic permeability distribution, at excitation frequencies between 1 Hz and 100 kHz. A similar numerical and experimental work concerning an EC sensor over a layered material has been performed by Augustyniak et al. [8]. These works allowed for physical interpretation of some key phenomena, such as e.g. “zerocrossing” frequencies due to the opposing influence of the conductivity and magnetisation, but they referred to a single, relatively simple configuration of the detection device and could not be easily generalised.
Only few researchers have aimed at constructing purely or dominantly analytical formulas capable of predicting and explaining the EC sensor output in the vicinity of a metal object. Among them, Kim et al. [9] introduce an equivalent network circuit representing the EC sensor and the ECs generated withing a metal plate. They start from the Biot–Savart’s formula, allowing determination of mutual inductance between the sensors’ coil and the EC flowing in the target object. The approach assumes, that the target object is flat, and axially symmetrical with regard to the sensor, which is rarely the case in the railway context, except for the examination of the separate axle or the wheel’s circumference. In another paper [10], Enoki et al. tackle the problem of predicting the resistance and inductivity variation of coils situated in the vicinity of a rail, at a high frequency typical for some railway sensors (450 kHz). They develop a closedform dependency of the electrotechnical parameters (resistance R, inductance L) on coil’s distance from the rail head top surface. Enoki’s approach has mixed analytical/empirical nature, which limits its application to a narrow set of rail/sensor configurations. Yet more importantly, it does not derive the results from the fundamental parameter, which is the space and time distribution of the magnetic induction. Instead, the skin effect and the associated field distribution are implicitly involved in a set of intermediate empirical parameters. Finally, Klein et al. [11] has recently defined a semianalytical model of the EC response of a coil operating inside two conducting concentric tubes. The excellent agreement was found between the model and the reference (numerical and experimental) data, but still the approach remains limited to a precisely defined sensor/object configuration.
To sum up this brief literature review focusing on the special area of axle counters, the experimental observations and advanced signal processing tend to be separated from theoretical background and magnetic field modelling, while the analytical approaches—useful as they are—can not be generalised to complex cases typical for industrial practice. More importantly, to the best authors’ knowledge, there have been no published attempts to precisely define a relationship between the magnetic field and electrotechnical variables in the railway EC sensor context.
Our approach aims at combining numerical and analytical strategies of representing the EC sensor setup, providing the key link between the magnetic field and the behaviour of the circuit, for an arbitrary rail/wheel/coil configuration, with as little calibration and empirical parameters as possible.
The approach starts from the fundamental observation that both the effective resistance R and inductivity L of the measuring coil are dependent on the presence of a metallic object. Starting from electrotechnical formulas relevant to either a serial or parallel resonant circuit, we postulate a relatively simple association between effective values of (R, L), and the amplitude/phase of the magnetic induction within the coil, which in turn can be determined from a relatively straightforward FEA. This is a significant step towards prediction of the output signal of any coilbased sensor in the vicinity of an arbitrarily shaped rail and wheel, and such an approach is potentially extensible to other areas of application of ECbased devices.

focusing on the transition between electromagnetic fields and the circuit electrodynamics, and proposing an original, physicsbased coupling formula, which allows for exhaustive explanation of experimental phenomena used in the commercial axle counters,

tacking the highfrequency domain (as compared with 50 Hz or 200 Hz in the aforementioned articles), where capturing skin effects with appropriate mesh size and shape is a difficult task,

multistage validation involving simple lab sample, then a short rail track segment, and finally a study and successful reproduction of service data of the actual measuring devices.
On the other hand, we have decided not to propose modifications of standard FE algorithms, and precise determination of material data (including nonlinearities) has been given secondary attention.
2 Basic Version of the Field–Circuit Coupling Function

the purely real characteristics of the magnetic flux causes a “normal” inductance of the coil, influencing uniquely the imaginary component of the circuit’s impedance,

any imaginary component of magnetic flux generates an “anomalous” contribution to the inductance, disturbing the real component of circuit’s impedance; this fact is fundamental for signal generation and interpretation in EC NDT.
The component − ωL_{ED} can also be termed R_{ED}, and represents the EC related contribution to the effective resistance of the circuit.
This Eq. (10) constitutes a key hypothetical relationship between the AC magnetic field within the sensor coil and the electrotechnical characteristics of a serial resonant circuit. Once validated, the presented set of equations allows for the at least qualitative prediction of the voltage–current function of the sensor coil, which is fundamental for construction and optimization of ECT measurement devices.
Note, that in the following discussion, the lefthand side of Eq. (10) is referred to as “ECARL” (inductancerelated), and the righthand side as “ECARB” (fieldrelated), where ECAR stands for “Eddy Current Activity Ratio”.
3 Case of a FeSi Plate
3.1 FeSi Plate: The Measuring Setup
In order to validate the basic version of the field–circuit coupling function, a fullycontrolled laboratory case of a thin flat FeSi plate has been studied.
The impedance analyser HIOKI35225 has been applied to record the following parameters: resistance, inductance, current/voltage phase shift and impedance modulus. The measurements have been performed at 100 kHz (the maximum frequency available for this analyser) with an imposed voltage range of 1 V. Four coaxial cables connected the analyser to a coil, which could either be placed “in the air” or in the vicinity of the studied object (a metal plate or the rail segment). The plate had dimensions: 160 × 100 mm and the thickness of 0.35 mm.
Characteristics of coils used in laboratory tests
Diameter (mm)  Axial length (mm)  Number of turns  

Coil X1  28  10  100 
Coil X2  16  10  100 
The X1 coil’s liftoff distance in this experiment was constant, set to 1 mm.
3.2 FeSi Plate: The Finite Element Model
The sinusoidally varying current density has been applied to the coil’s cross section, resulting in the total current of 1 Aturn.
The magnetic permeability of the plate has been adopted as 4000, and its resistivity set as 5 × 10^{−7} Ohm meter, which are values typical for 3% Si electrical steel.
The magnetic flux has been forced to be parallel to all the model’s boundaries, and the EC timeintegrated potential has been coupled at all the nodes belonging to the metal plate and lying on the symmetry planes, in order to assure the correct current flow at these boundaries.
3.3 FeSi Plate: Comparison of Experimental and Modelling Results
A single series of measurements of the coil’s resistance and inductance and the corresponding field computation has been performed for the configuration described in the Sect. 3.2, at the excitation frequency of 100 kHz.
Comparison of experimental and numerical results for the FeSi plate test
Parameters  Symbols  Values  Units 

Measured resistance “in the air”  R_{0}  9.0  Ω 
Measured resistance “by the plate”  R_{1}  20.9  Ω 
Measured inductance “in the air”  L_{0}  340  μH 
Measured inductance “by the plate”  L_{1(norm)}  336  μH 
Eddycurrent inductance component  L_{ED}  − 19  μH 
Shielding inductance component  L_{SH}  4  μH 
Averaged imaginary induction in the coil  B_{Im}  − 2.47  μT 
Averaged real induction in the coil  B_{Re}  43.4  μT 
Eddy current activity ratio deduced from measurement [L_{ED}/L_{1(norm)}]  ECARL  − 0.057  – 
Eddy current activity ratio deduced from simulation (B_{Im}/B_{Re})  ECARB  − 0.057  – 
The correlation between the measured and the simulated ECAR is perfect for the case of welldefined laboratory conditions, including a constant liftoff distance, simple object’s shape, and macroscopically flat, polished surface.
The next step towards the validation of the coupling function is studying a nontrivial geometry exhibiting a very rough surface and having uncertain material properties, at various coil’s liftoff distances.
4 Case of a NarrowGauge Rail
4.1 NarrowGauge Rail: Measuring Setup
4.2 NarrowGauge Rail: Finite Element Model
The magnetic permeability of the rail segment has been adopted as 200, and its resistivity set as 2 × 10^{−7} Ohm meter. These parameters were collected from [17, 18, 19], and accepted as typical for steel grades used for railway tracks.
The magnetic flux has been forced to be parallel to the symmetry plane, and the EC timeintegrated potential has been coupled at all the nodes belonging to the rail and lying on the symmetry planes, in order to assure the correct current flow at these boundaries.
The sinusoidally varying current density has been applied to the coil’s cross section, resulting in the total current of 1 Aturn.
4.3 NarrowGauge Rail: Comparison of Experimental and Modelling Results
The validation of the coupling function in the case of a rail track segment was carried out similarly to the case of a thin FeSi plate. The coil designated as X2 was used to measure the effective resistance and inductance at varying coil–rail distances.
 (a)
magnetic permeability and electric conductivity of the rail material were unknown and could only be roughly estimated,
 (b)
the object’s surface was pitted and covered with oxide/protective coating of unknown characteristics,
 (c)
the setup was not shielded against external electromagnetic noise.
It is interesting to note, that the ECAR at 1 mm (− 0.057) was found to be precisely equal to the one determined in the case of the FeSi plate, despite different coil characteristics and material properties. Decision, whether this correlation was due to some fundamental physical law, or obtained accidentally, would require additional tests for other sample/coil configurations.
5 Case of a commercial Electromagnetic Axle Counter
The postulated transfer function provided an ideal agreement in a fullcontrolled FeSi plate case, and a good agreement in the narrowgauge rail track case, where some uncertainty factors (esp. surface roughness) have been allowed.
An ultimate test of a fully industrial situation has been performed, in which the output from a multicoil sensor was recorded while train wheels moved in its vicinity. The device has been developed and commercialised by VOESTALPINE (Sopot, Poland) under the brand UniAS2. It is supposed to record two key outputs (dynamic resistivity R_{d}, and the resonance frequency ω_{r}), directly derivable from the amplitude and phase relationship between the current imposed on the circuit and the resulting voltage. The goal of the test was to deduce these key outputs based on the magnetic field computation and the analytical coupling function, and to compare them against experimental ones.
5.1 Axle Counter: Finite Element Model
Similarly to the narrowgauge rail case, the meshing in the subsurface zone was generated with a progressive bias, so that the outermost layer featured thicknesses of 1/100 mm, suitable for correctly reproducing the skin depth effect. Only a 120° sector of the wheel was modelled for the sake of reduction of the number of elements and associated analysis time. The device was modelled in details, including all the metal plates, screws and bolts, and their surfaces were meshed in a manner adequate to capture the EC influence.
5.2 Field–Circuit Coupling Function for a Parallel Resonant Circuit

the effect of wheel velocity is negligible, because of the relatively high operating frequency of the device, exceeding 100 kHz, as compared with the frequency of the wheel passages (this could be false for high speed trains, and would require separate study),

only two extreme cases are considered: “NW”—NoWheel, and “WP”—WheelPassing, with the wheel axis situated precisely over the geometrical center of the device,

the nonlinearity of the rail/wheel material B(H) curve can be neglected,

single “effective” L, R and C values can be used, being the arithmetic sum of components (coils, capacitors, resistors) placed on all branches of the parallel circuit,

the magnetic field amplitude and phase in the neighbouring coils can be averaged and replaced by a single “effective” B composed of a real and an imaginary part.
Note, that the formulas (18) and (20) are exactly equivalent to (10), while the additional formula (19) constitutes a means of predicting the “WheelPassing” inductance of the circuit from the “NoWheel” inductance and the ratio of calculated magnetic fields.
Knowing the effective L and its EC related component L_{ED} from the formulas above one may now determine the key outputs R_{d} and ω_{r} of the circuit for the “WheelPassing” state, and comparing them against the reference “NoWheel” situation. This is fundamental from the point of view of the calibration and interpretation of the signals from the device. Actually, the same universal procedure allows for determining the dynamics of the device for any combination of the rail and wheel, including variable geometry, material data, operating frequency.
5.3 Axle Counter: Comparison of Experimental and Modelling Results
Validation of the presented formulas consisted in calculating the magnetic induction within the coils in both the NW and WP situations, applying the coupling functions and finally comparing the “predicted” and “measured” device dynamics.
The comparison of simulationbased key outputs against averaged data gathered from service is presented below.
Comparison of key outputs derived from modelling and from experiment in the axle counter test
Modelling  Averaged inservice measurements 

Change of dynamic resistance R_{d} during wheel rollon  
− 32%  − 40% 
Change of resonant frequency of the circuit during wheel rollon  
+ 700 Hz  + 1000 Hz 
In the commercial continuation of the presented research (detailed results not presented here), both the resonance curve exhibited by the sensor and its maximum output impedance were correctly predicted by the simulation linked the circuit by analytical coupling functions. These results have served as a starting point for a series of modelling of various rail and wheel configurations, and provided a deeper understanding of signal dynamics for three variants of the wheel proximity sensors produced by VOESTALPINE AG.
6 Conclusions
In this paper a concise and fairly simple, yet universal methodology for predicting some output characteristics of an EC sensor has been presented. The algorithm in its basic version requires no empirical parameters, relying exclusively on design parameters of the measuring device and the studied object (shape, material, frequency, structure of the electric circuit). The algorithm starts with a standard linear harmonic FEA determining the magnetic induction within the coil(s), followed by the application of coupling functions, linking the field variable to the electrotechnical variables of the circuit. The method finally allows for prediction of key parameter named ECAR, involving the variation of coil’s inductance due to the proximity of metal objects.
The method has been validated in three situations of gradually increasing complexity. An ideal agreement between the predicted and experimental results has been achieved in the fullycontrolled laboratory tests. A more demanding case of a rail segment with a rough surface produced a very good correlation. Finally, the proposed algorithm passed the fully industrial test, in which a commercial UniAS2 device composed of four coils, placed against a rail and a passing wheel, produced outputs which were successfully predicted by the FE calculation coupled with the coupling functions appropriate for the parallel RLC circuit.
As compared to the fully numerical solution (available to e.g. Flux/Altair software), the presented approach has two advantages. First, the FEA solution required here is linear, whereas the direct computation of the coupling between the magnetic field and the circuit is nonlinear and more difficult to setup. Second, the analytical passage between the magnetic field and the circuit variables provide a unique insight into the physics behind the device’s behaviour and guides changes in the design e.g. by optimal positioning of the sensor.
Proving the validity of the formulas defining ECAR and the capability of predicting sensor’s output changes during the wheel passage are a significant, but only partial progress, because of the relative (not absolute) character of the determined relationships. A complete success shall be attained once direct formulas for both L_{ED}(B) and L_{SH}(B) are found and validated. This is expected to provide a truly universal and practical methodology for predicting, understanding and optimising characteristics of new EC NDT devices and sensors.
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