Static or slowly varying magnetic fields can affect the performances of linear variable differential transformer by inducing a position reading drift. The problem is barely addressed in LVDTs' datasheets, and no quantitative information on the induced error is given. An LVDT finite element model is here presented together with its experimental validation in order to propose a tool for the study of the effects of external magnetic fields on LVDTs and for the design of less sensitive devices. The LVDT model has been validated in standard working conditions and in presence of an external magnetic field by means of a complete set of experimental measurements performed on a custom prototype, manufactured following the FEM guidelines.
The linear variable differential transformer is a magnetic position transducer. Its use is very widespread due to its features of being a contactless sensor, with virtually infinite resolution and high accuracy [
The LVDT sensor is basically a transformer with one primary winding, in the center of the cylindrical structure, and two secondary windings, one of each side of the primary, wound on a cylindrical support (Figure
Typical LVDT layout and working principle (longitudinal section).
Although the LVDT reading accuracy can be guaranteed even in critical and noisy installations [
FEM-aided analysis allows a deeper investigation regarding local variables and physical magnitudes, such as local values of magnetic flux density or magnetic field [
The results coming from the FE analysis are then validated on a customized LVDT prototype, whose structure closely reflects the one of the finite elements model itself. A complete set of measurements has been performed on the manufactured prototype, in standard working conditions and in presence of an external interfering magnetic field, in order to verify the agreement with the model. Since the LVDT can be supplied by either voltage [
In Section
An FEM model for the simulation of the LVDT sensor has been developed using the simulation software FLUX. This simulator is particularly suited for the finite element analysis of electromagnetic problems involving 2D and 3D geometries [
The purpose of such an FEM analysis is to conceive a model of linear variable differential transformer which can be used as a tool for analysis and design of LVDT exposed to external magnetic fields. The availability of such a model would allow an immediate feedback in the analytical study of the physical phenomenon of the external magnetic fields influence on LVDT reading, as well as in the design process of an LVDT-like structure with immunity to external magnetic fields.
Being in principle the external field not known in many applications (neither in terms of amplitude, nor in terms of time evolution), a recalibration of the device in presence of interfering field, which could be an effective solution for purely DC fields, cannot be of help in this case. Furthermore, the external field may not be always present. Indeed, in presence of a slowly varying external field (i.e., the frequency content of such a field being in the ultralow frequency range whereas the internal LVDT magnetic field harmonics are in the range of few kHz), at each reading the LVDT will experience an error which is a function of the value of the interfering field (which is nearly constant). Therefore, at each reading, the effect on the position can be seen as due to a DC external field. However, the existence of an intrinsic time variation of such a field, for what has been said, invalidates the possibility to recalibrate the device.
The LVDT geometry presents cylindrical symmetry. On the other hand, an interfering magnetic field impinging the LVDT structure can be in principle arbitrarily oriented. However, an arbitrarily oriented magnetic field can be seen as the superposition of longitudinal (parallel to the LVDT axis) and transversal (perpendicular to the axis) components. Given that such sensor is more sensitive to longitudinal magnetic fields [
The simulated LVDT model slightly differs from the simple structure presented in Section
2D longitudinal scheme of the LVDT sensor FEM model (not in scale).
The structure is enclosed in a ferromagnetic cylindrical case with two end caps. The ferromagnetic case, together with the end caps, has two main functions: it closes the LVDT magnetic circuit and acts like a first shielding against external magnetic fields. The core is a cylinder whose length is equal to the secondary coils length.
Given the cylindrical symmetry, the simulation geometry takes into account only half of the longitudinal section of the sensor (Figure
2D simulation geometry and 3D reconstruction. In the infinite box, a geometrical transformation is performed in order to simulate the infinite space [
The structure has a high aspect ratio, thus a fine mesh has been chosen in order to discretize the thicknesses whereas the mesh along the length of the sensor can be coarser. By doing so, the mesh has been optimized using triangular elements on all the geometry. The meshing and the solving parameters for the geometry are reported in Table
Mesh and solving information.
Info | Parameters |
---|---|
Mesh type | 2D triangular mesh |
Mesh order and density | 2nd order, nonuniform |
General meshing rule | 3-4 elements per thickness |
Number of poor elements | 2.8% |
Number of nodes | About 43000 |
Solving scenario | Time transient |
Target relative epsilon of the Newton-Raphson solver | 10−4 |
Maximum number of iterations | 100 |
Solving time (simple computation: no parameters except time) | 24 minutes |
The FEM modeling of magnetic materials properties for the study of the LVDT magnetic interference is not trivial. As a matter of fact, given the presence of an external magnetic field superposed to the one due to the LVDT working principle, the materials magnetic properties have to be correctly described in all regions of the BH plane for the simulations to closely match their actual behavior. Unfortunately, FEM simulators model the nonlinearity of magnetic materials by considering their normal magnetization curve, without taking into account major and minor hysteresis effects [
For these reasons, the materials which have been chosen for the magnetic parts of the LVDT sensor model had been through an annealing procedure. The preference of having annealed samples instead of not treated materials lies in the fact that in annealed state such samples exhibit high permeability and narrow hysteresis cycle [
PERMENORM 5000 H2 (50% Ni-Fe alloy) has been chosen as core material whereas ferromagnetic steel has been used for the cylindrical case, since these materials were available to be used and they had been through an annealing procedure.
Regarding the coils’ specifications, a number of turns of 1500 has been chosen for the primary winding, in order to have a sufficiently high field inside the sensor whereas a number of turns of 1600 has been chosen for both secondary coils, setting the transformation ratio.
The time transient solver has been chosen. The Newton-Raphson method is used for nonlinear solving and a maximum number of 100 iterations is set for each time instant. The relaxation factors for this algorithm are calculated through the Fujiwara method [
Geometrical distances, including crucial parameters (e.g., the core displacement) have been parameterized in order to allow rapid parametric simulations embedded in the same simulation scenario. The numerical transients have been avoided by using an initialization by static computation [
The parameterization of the core displacement allows performing a set of simulations at different core positions. For each of them, the geometry is remeshed and the time transient computation performed [
For the interfering field simulation, taking critical installations as an example [
The simulation results for voltage and current supply, in presence and in absence of the interfering magnetic field, are here shown.
Although the two supply modes are both used in commercial LVDTs, the need of studying the phenomenon with two different supplies comes from the fact that when the supply signal is a purely sinusoidal current, the overall magnetic field is sinusoidal whereas with voltage supply the overall magnetic field is distorted due to the nonlinearity of the magnetic media. Hence, the influence of the external magnetic field can, in principle, play different roles in the two examined cases.
For each supply case, the LVDT characteristic curve, which shows the first harmonics of the secondary voltages with respect to the reference position, is shown. In addition, the ratiometric has been computed as follows:
In case of interference, the external magnetic field yields an error on the measured position. Hence, a position drift has been computed, at each core position, as the difference between the position read in presence of the external interference and the position read in absence of the interference. In both cases, the position is read using the ratiometric reading technique, by means of the calibration curve obtained in standard working conditions. Actually the position drift allows performing a relative study (i.e., a study on a position variation) in order to highlight the effect of the external magnetic field on a certain position.
The sensor has been supplied by a 3.5 V peak sinusoidal voltage at 2000 Hz. The characteristic curve in this case is displayed in Figure
Simulations and measurements results regarding the LVDT secondary characteristic curve (voltage supply). The measurement data are depicted with the corresponding uncertainty.
It can be noticed that the two curves have a dual behavior in terms of trend, which recalls the working principle of LVDT sensors. The curve is symmetric with respect to the null position, due to the complete symmetry of the device. The simulations show that the secondary transformation ratio of the sensor goes from 0.54 (minimum core coupling) to 1.66 (maximum core coupling).
The behavior of the ratiometric as a function of the position has been evaluated, and the relative results are depicted in Figure
Simulations and measurements results regarding the ratiometric (voltage supply). The measurement data are depicted with the corresponding uncertainty.
In simulation, the nonlinearity of the device resulted, 0.7% in the range (−20 mm, 20 mm]. The linearity becomes more acceptable when the core position range is reduced: nonlinearity error is 0.11% in the range [−10 mm, +10 mm].
The results for the longitudinal interference case are displayed in Figure
The sensor has been fed with a current sine wave at 2 kHz whose amplitude, 24.0 mA, has been chosen in order to have the amplitude of the first harmonic of the primary voltage of about 3.5 V when the core is in null position. By doing so, the simulation results can directly be compared with the ones obtained with voltage supply.
The characteristic curve in the current supply case is displayed in Figure
Simulations and measurements results regarding the LVDT secondary characteristic curve (current supply). The measurement data are depicted with the corresponding uncertainty.
Again, a dual behavior of the two curves in terms of trend and the symmetry around the null position can be noticed. In this case, the secondary transformation ratio of the sensor goes from 0.54 (minimum core coupling) to 1.66 (maximum core coupling).
Figure
Simulations and measurements results regarding the ratiometric (current supply). The measurement data are depicted with the corresponding uncertainty.
Simulations and measurements results regarding the position drift (voltage supply). The measurement data are depicted with the corresponding uncertainty.
The results regarding the position drift are depicted in Figure
Simulations and measurements results regarding the position drift (current supply). The measurement data are depicted with the corresponding uncertainty.
The simulation work constituted the starting point for an LVDT prototype manufacturing. The dimensions, materials, geometry, and coils specifications reflect the simulation criteria. As already said in the last paragraph, all magnetic materials have been through an annealing process.
The primary coil has been wound on 2 layers with a wire diameter of 0.28 mm (in order to allow possible measurements even with high currents) whereas the secondaries are single layer coils with a wire diameter of 0.06 mm, since they are supposed to be connected to high impedances, typical of DAQs. An expanded view of the manufactured prototype, with highlighted information, is displayed in Figure
Expanded view of the custom prototype used to validate the FEM model. Dimensions scale is in centimeters.
A complete automatic test bench, whose details can be found in [
The external longitudinal magnetic field is generated by means of an external calibrated solenoid, fed by a DC current so as to have the desired amplitude and a uniform distribution for the field along the solenoid length. In addition, a demagnetization procedure has been foreseen in the measurement process [
The measurements in absence of external magnetic field constitute the calibration of the prototype, yielding to its calibration curve. In presence of external field, the position is measured by using this curve, and the position drift is computed as the difference of the position measured in presence of external field and that measured in absence of it. As for the simulations, the position drift analysis is meant to be a relative study whose function is to point out the effect of the external magnetic interference.
In all experimental results, the related expanded measurement uncertainty is depicted too. This has been calculated on 30 repeated measurements, supposing the values spread as a Gaussian distribution in the measurement interval. A Chi-square test on the repeated electric zero research results confirmed that the null point of the LVDT prototype is spread as a Gaussian distribution with a standard deviation of 6 micrometers. The uncertainty of the measurements at different positions is, therefore, dominated by this component, since the other sources of uncertainty coming from the test bench give much smaller values [
The uncertainty on the position drift has been calculated as follows:
The prototype characteristic curve is depicted in Figure
Regarding the voltage amplitudes, the agreement between simulation and measurements results is good, since it is always greater than 85% whereas it increases to more than 95% when reducing to (−10 mm, +10 mm) the core position range.
The results for the ratiometric are depicted in Figure
Regarding the interference conditions, the related position drift is presented in Figure
A remark on the measurement uncertainty has to be done. In Figure
Summary of experimental results.
Item | Voltage supply | Current supply |
---|---|---|
Feeding signal amplitude | 3.5 V ( | 23.0 mA ( |
Transformation ratio | from 0.48 ( | from 0.48 ( |
Ratiometric range | from −0.58 (− | from −0.57 (− |
Voltage swing (V) | 4.30 ( | 4.35 ( |
Non linearity error (%) | 1.11 ( | 1.12 ( |
Primary voltage repeatability (mV) | 0.8 (no interference) | 1.5 (no interference) |
Secondary voltage repeatability (mV) | 1.6 (no interference) | 2.1 (no interference) |
Measured position uncertainty (micrometers) | 11 (no interference) | 13 (no interference) |
Position drift values (micrometers) | from 71 ( | from 58 ( |
Position drift uncertainty (micrometers) | 19 | 55 |
Agreement with simulations (characteristic curve) (%) | >85 in (−20 mm, 20 mm) | >86 in (−20 mm, 20 mm) |
Agreement with simulations (ratiometric) (%) | >83 | >84 |
The measurements in current supply have been performed by using a sinusoidal feeding current whose peak amplitude is 23.0 mA. This value ensures a first harmonic amplitude of the primary voltage of 3.5 V when the core is in null position. This current value is in agreement with the one coming from the simulations; thus, there is a good match between measurements and simulations, regarding the primary impedance.
Figure
The behavior of the ratiometric values with respect to the core position is shown in Figure
Concerning the interference case, the results regarding the position drift are depicted in Figure
Furthermore, a remark on this agreement in current supply has to be done. The secondary voltage of an LVDT can be expressed as [
Regarding the measurement uncertainty on the position drift, in this case it is higher than the correspondent with voltage supply. Actually, with current supply the uncertainty on the primary voltage is higher (the primary voltage is not fixed by the generator), as reported in Table
For a complete overview of the results in different supply cases and the comparison with simulations, see Table
The finite element analysis is used in this work to conceive a FEM model of LVDT for the study of the effect of magnetic interference on the sensor. The model is presented together with simulation results of the LVDT characteristic curve and the effect of a 1 mT external flux density on the position reading. A custom LVDT prototype has been manufactured in order to validate the model, by means of an automated measurement test bench [
The scientific target of the proposed FEM model and the simulation procedure described in standard working conditions and with external interference is to make available a tool both for the study of the physical phenomenon and for the design of future LVDTs where the sensitivity to external field can be correctly taken into account from the very preliminary phase.
The future design of an LVDT with less sensitivity to external magnetic fields could indeed profit from considerations and results based on the FEM model here presented: for example, the simulation data can affect the improvement of the performances of an LVDT by giving rapid design feedbacks regarding the choice of a certain material for the core or for an external magnetic shield, or allowing a sensitivity study with different coiling specifications, magnetic field intensity and orientation, and so forth. Furthermore, the proposed model and its future expansion can significantly support the study (analytical or FEM aided) of a more complex and general case of magnetic interference on linear variable differential transformers (e.g., considering nonuniform and slowly varying interfering magnetic fields).
The authors would like to thank engineer Antonio Pierno and Dr. Daho Taghezout (applied Magnetics) for the support on measurements and simulations respectively. They would also like to thank Professor Vittorio Giorgio Vaccaro, Professor Felice Cennamo and Professor Yves Perriard for their useful discussions and suggestions.