Abstracts: The performance of an Eddy Current Displacement Sensor (ECDS)
can be optimized by changing the factors of the sensor coil, such as geometry,
material, and working frequency. Finite Element Method (FEM) can precisely
simulate the sensor, its target, and the conditions in which the sensor is working. It
is important to study the ability of FEM in the simulation of eddy current sensors for
displacement measurement. In this paper, we simulated an eddy current sensor coil
and investigated its impedance response in different scenarios using Ansys software.
The experimental findings were consistent with the simulation results. For operating
frequency at 180 kHz, comparison between simulation results and experimental
using LCR meter and LDC1000EVM module results, the mean absolute percentage
error were about 0.75% and 0.6%, respectively.

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3D FINITE ELEMENT SIMULATION AND EXPERIMENTAL
VALIDATION OF EDDY CURRENT DISPLACEMENT SENSOR
Hoang Si Hong
1*
, Nguyen Van Tien
1
, Cung Thanh Long
1
, Nguyen Van Dua
2
Abstracts: The performance of an Eddy Current Displacement Sensor (ECDS)
can be optimized by changing the factors of the sensor coil, such as geometry,
material, and working frequency. Finite Element Method (FEM) can precisely
simulate the sensor, its target, and the conditions in which the sensor is working. It
is important to study the ability of FEM in the simulation of eddy current sensors for
displacement measurement. In this paper, we simulated an eddy current sensor coil
and investigated its impedance response in different scenarios using Ansys software.
The experimental findings were consistent with the simulation results. For operating
frequency at 180 kHz, comparison between simulation results and experimental
using LCR meter and LDC1000EVM module results, the mean absolute percentage
error were about 0.75% and 0.6%, respectively.
Keywords: Displacement sensor; Eddy current sensor (ECS); Finite element method (FEM); 3D model.
1. INTRODUCTION
Non-contact displacement sensors increasingly contribute to the solution of demanding
measurement tasks in which the sensors must operate without contact with the
measurement targets. Depending on the applications, the most common types of non-
contact displacement sensors are laser triangulation, capacitive, and eddy current [1]. In
measurement applications of small distance with a conductive target, eddy current sensors
are preferred due to their compact size, high resolution, excellent temperature stability, and
contamination resistance. One good example of ECDS applications is position
measurement of a levitated target in magnetic bearings [2]. Eddy current sensors with sub-
nanometer resolution, ultra-stable [3], extended linear range [4, 5], and high-temperature
resilience [6] have been researched.
Finite Element Method (FEM) has been a prominent method in ECDS research since
the 2000s due to its growingly powerful ability. Y. Lai utilized FEM in the development
and optimization of an ECDS for engine monitoring [6]. FEM was used to develop ECDS
with extended measurement range [4, 5, 7] and improved resolution and temperature
stabilization [3] using different methods such as using new coil material, different probe
structures or compensation circuit. In recent years, while FEM is mainly used in
researches of eddy current sensor for inspection applications such as in a study by Long.
TC [8], it is also common in other applications such as measurement of rotation speed [9].
However, only a small number of researches has been done to study FEM’s ability in
ECDS development [10–12]. This paper will focus on studying the finite element method
in ECDS simulation with experimental validation.
A. Principles and behavior of eddy current sensor
Figure 1 shows the working principle of an Eddy Current Sensor. When an AC current
flows inside a sensor coil is placed in the vicinity of a conductive surface, the oscillating
magnetic field of the coil will induce eddy currents inside that surface. The secondary
magnetic field created by eddy currents opposes the initial one, causing the magnetic flux
and therefore the inductance of the coil to decrease. Meanwhile, the eddy current also
dissipates energy which leads to the increment of the coil’s resistance.
The quality factor Q is defined as in equation (1).
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(1)
where ω is the operating frequency of the sensor in radians per second.
The higher value of Q, the more purely reactive the sensor which leads to high accuracy
and stability [13]. Inductance L, resistance R, impedance Z, and quality factor Q are
dependent on many factors including relative distance between sensor coil and the target (or
standoff) x, working frequency f, resistivity ρ, and permeability µ of the target [14].
Figure 1. Eddy current sensor working principle.
When a nonmagnetic target approaches the coil, the inductance decreases while the
series resistance increases. Typically, the changes in the series resistance are negligible
compared to the sensor inductive impedance variations as the target moves. While
resistance R is highly sensitive to temperature, target conductivity, and frequency
variation, the coil’s inductance could be very stable to these factors when the frequency f
or the target conductivity is large enough [3, 14]. Because of the mentioned reasons, the
coil’s resistance is often neglected in displacement measurement. Depends on the
modulation techniques, the performance of an ECDS can be quantified by the relationship
between either the coil’s impedance Z or its inductance L and the target position x. In eddy
current displacement measurement, high frequency is usually preferred to maximize Q.
However, the working frequency cannot access and must remain at least a factor of three
below self-resonant frequency (SRF) [13] which is given by
(2)
where CP is the parasitic capacitance of the coil.
Thus, printed coils are sometimes preferred because of their smaller parasitic
capacitance compared to wire-wound coils. Besides the mentioned factors, the coil’s
impedance is also affected by coil geometry (outer diameter, inner diameter, and
thickness), target geometry (such as size, flatness, and thickness [13]). The guidelines for
designing a near-optimal coil have been studied in [12, 13, 15].
B. Research overview
In this paper, we first simulated a printed PCB coil. The relationship between the
parameters of the coil and its working frequency has been obtained in two cases: the coil
in isolation and the coil in proximity to a conductive target. An optimal working frequency
range of the coil was concluded from the simulation results. The coil then simulated at
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different standoffs to an ideal aluminum target using Ansys Maxwell. An experiment has
been done with the real coil and an aluminum target using a commercial evaluation
module by Texas Instruments and an LCR meter. The results show that the relationship
between the coil’s normalized inductance and its standoff to the aluminum target obtained
by simulation and experiment are identical. Figure 2 shows the printed coil used in this
paper which parameters were shown in Table 1 below.
Figure 2. Two-layers PCB coil used in the simulation and experiments.
Table 1. Coil design parameters.
Parameter Value
Sensor diameter 551 mil (14 mm)
Number of layers 2 layers with 62 mil (1.8mm) PCB thickness
Thickness of PCB copper 1 Oz-Cu (35µm)
Number of turns 23 (each layer)
Trace thickness 4 mil (0.102mm)
Trace spacing 4 mil (0.102mm)
2. SIMULATION AND DISCUSSION
A. Simulating coil in isolation
First, we simulated the isolated coil and analyzed its DC parameters including
inductance LDC, resistance RDC, and the parasitic capacitance CP. Ansys Q3D Extractor was
used to simulate a detailed 3D model of the two-layered PCB coil and extract its LDC, RDC,
and CP values. Figure 3(a)&(b) show the 3D model of the PCB coil and the meshing result
in Q3D Extractor.
Figure 3. Detailed & Simplified 3D models of the PCB coil
with Q3D Extractor & Maxwell.
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We also simulated another simplified model of the coil, placed in an air region with
Maxwell for comparison. As shown in Figure 3(c), the simplified 3D model consists of
two disc-shaped copper layers with the same parameters, such as outer and inner diameters
and copper thickness as the printed coil. The 3D model was placed inside a large air region
and excited with AC current.
The simulation results of the detailed model and the simplified one were compared in
Table 2. The resistive value calculated with the simplified model was multiplied by two to
compensate for the effects when neglecting the air gaps between coil turns.
Table 2. Comparison of results obtained from detailed and simplified models.
Parameters Q3D Maxwell Error (%)
L 16.092 µH 15.844 µH 1.54
R 6.484 Ω 6.372 Ω 1.73
C 0.648 pF x x
From the obtained values, the self-resonant frequency (SRF) of the coil can be
calculated using equation (2) . To study the effects of working frequency
variation around the self-resonant frequency value, the 3D model was analyzed with
exciting frequency varied from 100kHz to 100MHz. Figure 4 shows the relationship
between the parameters of the isolated coil and its exciting frequency. As shown in Figure
4(a), the equivalent serial resistance reaches its highest value at SRF and rapidly plummets
as frequency increases after SRF value. Meanwhile, the equivalent serial inductance value
reaches its peak right before falling to zero as frequency reaches SRF value. When the
frequency is higher than self-resonant frequency, the coil will no longer behave like an
inductor. Quality factor Q value reaches its peak around 20MHz then swiftly decreases to
zero at SRF, as shown in Figure 4(b). In conclusion, the operating frequency of the coil
should be kept below 10MHz to limit the parasitic capacitive effects.
Figure 4. Relationship between parameters of the isolated coil and exciting frequency (a)
Inductance L (µH) and resistance R (Ω); (b) Quality factor Q.
B. Simulating coil close to a conductive target
In the simulation, the coil was placed with its axis is perpendicular to the surface of an
aluminum target with the standoff is a variable value. The size of the target has been
chosen to not have any interference with the results. The thickness of the target is larger
than 3 times of calculated standard skin depth at the lowest exiting frequency; the size of
the target is set to be larger than three times of the coil’s diameter. Due to the symmetry of
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H. S. Hong, , N. V. Dua, “3D finite element Eddy Current Displacement Sensor.” 138
the geometry to the XZ and YZ planes, the model, including the coil, the target, and region
can be simplified to ¼ portion of the original model. The simplified model gives the
results with ¼ smaller values compared to the initial one. Maxwell provides boundary
conditions which support the simplification of symmetric models, but in this case, the
default boundary condition on XZ and YZ planes were sufficient. Ansys Maxwell3D
provides six different solution types, designed to solve Maxwell’s equations within the
scope of the specific solution type. In this paper, we used Eddy Current solver because it
solves sinusoidally-varying magnetic fields and induced fields such as eddy currents.
Figure 5. Simulation of the coil and an aluminum target.
The model consists of ¼ of the coil and ¼ of the target is shown in Figure 5. The model
was placed inside an air region shown in Figure 5(a). The region should be large enough
that fields have largely diminished at the boundary of the region. In this case of modeling
an unshielded coil, the region should be around 20 times bigger than the objects’ size. The
cutting planes of the model touch the background allows excitations can be assigned to the
coil. The two layers of the coil were excited with “stranded” currents shown in Figure
5(c). A matrix and the number of coil turns were later assigned for calculation. The
meshing results of the coil and the target were shown in Figure 5(d) and(e). Note that the
surface closed to the coil of the target was assigned with a skin-depth mesh to reduce the
number of elements.
Figure 6. Normalized inductance with frequency varied
from 100Hz to 10MHz (at 1mm and 3mm).
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First, we studied the effects of exciting frequency variation on the parameters of the
coil. We simulated the coil with exciting frequency varied from 100Hz to 10MHz and
liftoff to the target is 1mm and 3mm, respectively. The normalized inductance values of
the coil at different frequencies were shown in Figure 6.
The results show that the variation of the inductance is more significant when the coil
is closer to the target. When the exciting frequency decrease, the inductance of the coil
increase toward the value of the isolated coil. When the working frequency is high
enough (around several MHz), the variation of coil inductance is neglectable. Following
the earlier conclusion, the optimal range of operating frequency should be from several
MHz to under 10MHz.
After the frequency variation effects on the inductance of the coil were analyzed, we
studied the effect of standoff variation on the coil’s parameters. The coil was simulated
with standoffs to the target varied from 1mm to 14mm. The obtained results shown in
Figure 7 were as expected of a typical eddy current displacement sensor coil. As the sensor
coil was moving away from the target, its sensitivity rapidly decreased, this limits the
measurement range of the sensor to less than 4mm. The sensor’s small measurement range
relate to its outer diameter can be explained by the shape of the coil.
Figure 7. Inductance (µH) with standoffs varied
from 1mm to 14mm (at 180kHz and 3MHz).
3. EXPERIMENTS AND DISCUSSION
Figure 8. Setup for experiments. Figure 9. (a) Evaluation module
LDC1000EVM; (b) Principles.
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Figure 8 shows a picture of the setup used for experiments. The sensor coil was
mounted in parallel with the surface of a flat aluminum target. The distance between the
coil and the aluminum target can be adjusted and measured.
An evaluation module LDC1000EVM (Figure 9(a)) by Texas Instruments was used in
this paper. The module consists of a parallel LC tank, an inductance-to-digital converter
LDC1000, and a microprocessor to communicate with computers via a USB cable. The
module working principle was shown in Figure 9(b), the sensor coil is kept at resonant
with a capacitor and the oscillation amplitude is regulated at a constant level. LDC1000
measures the equivalent parallel impedance RP and the resonant frequency of the LC tank.
The inductance value can be either calculated from the resonant frequency of LC tank fres
given in equation (3) or the equivalent parallel impedance RP given in equation (4) [16].
(3)
(4)
To measure the impedance of the coil, two different setups were used. In the first setup,
the impedance of the coils was measured with an LCR meter as shown in Figure
10(a)&(b). The coils were excited by a sinusoidal voltage with a magnitude of 1.5V and an
exciting frequency of 180kHz. With the exciting frequency is constant, this setup is similar
to an amplitude modulation (AM) circuit design. In the second setup, the impedance of the
coils was measured with the evaluation module LDC1000EVM and a computer as shown
in Figure 10(c)&(d). The working principle of the evaluation module LDC1000EVM is as
of a frequency modulation (FM) circuit design.
Figure 10. (a) Setup with LCR meter; (b) Setup with evaluation module; (c) A picture of
measurement with LCR meter; (d) A picture of measurement with evaluation module and
computer.
The coil’s impedance in simulations are smaller compared to the measured values using
LCR meter in experiments, as shown in table 3.
The inductance of the coil was measured with liftoff to the target varied from 1mm to
6mm in 0.5mm steps. The working frequency of the sensor with an evaluation module
LDC1000EVM varied from 3MHz to 5MHz when liftoff changes between 6mm to 1mm.
Because the effects of frequency variation can be neglected at high frequency, a constant
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frequency at 3MHz was used in the simulation for comparison with experimental results of
the evaluation module LDC1000EVM.
Table 3. Comparison between experimental and simulation vaule of coil's impedance
without conductive targets at 180kHz.
Parameters Maxwell Q3D LCR meter
Inductance µH 15.844 µH 16.092 µH 17.407 µH
Resistance Ω 6.372 Ω 6.484 Ω 10.332 Ω
Figure 11. Comparison between experimental and simulation results (a) LCR meter
results at 180kHz; (b) LDC1000EVM module results.
The comparison between the experimental results and the simulation results of
normalized inductance in the range of 1mm to 6mm was shown in Figure 11. Despite the
deviations in the values of coil’s impedance, the experimental results of normalized
inductance of the coil were consistent with the simulation figures. The mean absolute
percentage error (MAPE) between the experimental findings and simulation results in two
cases of using LCR meter and evaluation module were 0.75% and 0.6%, respectively.
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4. CONCLUSION
In this paper, we studied the Finite Element Method for simulating Eddy Current
Displacement sensors. The inductance response of the coil at different frequencies was
investigated in two cases: the coil in isolation and the coil closed to a conductive target.
The inductance of the sensor coil at different standoffs to an aluminum target was
measured in experiments and simulation. The archived results show that the measurement
range of the sensor is limited in the range of around 0mm to 3mm. At distances larger than
3mm, the sensitivity of the sensor is significantly decreased. The calculation and
optimization of sensitivity and measurement range are reserved for future works. In later
researches, the sensor can be examined at a better resolution, a smaller displacement
measurement can be archived. A complete ECDS system used for an exclusive task can be
designed and researched. This research shows the possibility of utilizing FEM in designing
and optimization of precision ECDS systems in the future.
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