A novel compensation algorithm for thickness measurement immune to lift-off variations using eddy current method
Mingyang Lu, Liyuan Yin,Anthony Peyton, Wuliang Yin
School of Electrical and Electronic Engineering, University of Manchester, M60 1QD, UK
Email:
Phone: +44 (0)161 3062885
Liyuan Yin
Kunming Science and Technology University
Abstract – Lift-off variation causes errors in eddy current measurement of metallic plate thickness. In this paper, we have developed an algorithm which can compensate this variation and produce an index that is linked to the thickness but virtually independent of lift-off. This index, termed as the compensated peak frequency, can be obtained from the measured multi-frequency inductance spectral data using the algorithm we developed in this paper. This method has been derived through mathematical manipulation and verified by both simulation and experimental data. Accuracy in thickness measurements at different lift-off proved to be within 2%.
Keywords: New compensation algorithm, eddy current testing, lift-off variation, thickness measurement
1. INTRODUCTION
The thickness of metallic plates can be measured by using both multi-frequency and pulse eddy current testing methods [1-11]. However, both methods suffer from errors caused by the so called lift-off effect. To address this issue, researchers have investigated a range of methods such as using different signal processing, feature extraction [12-13], sensor structure [14-16] and detection principles [17-20]. Multi-frequency eddy current sensing in the context of non-destructive testing applicationshas been the focus of the authors’ research in recent years. Conductivity and permeability depth profiling [21-22], non-contact microstructure monitoring [23-29] have been explored.
In a recent development [30], we proposed a novel design of an eddy current sensor, composed of 3 coils and operating as an axial gradiometer interrogated witha multi-frequency waveform. The difference in the peak frequencies of the impedance / inductancespectra from the gradiometerwas used for thickness evaluation and shown to be virtually immune to lift-off variations.
In this paper, we consider a simplerstructure consisting of just one coil pair. This method has the advantages of a less complicated mechanical configuration as well as improved accuracy, and avoids the need for a precise magnetic balance, as in the case of the gradiometer. The new technique exploitstwo facts. First, the peak frequency of the inductance spectral signal decreases with increased lift-off, and second, the magnitude of the signal decreases with increased lift-off.An algorithm has been proposed to compensate the change of the peak frequency. Theoretically, simulation and experiments show that the compensated peak frequency is nearly lift-off independent and therefore provide accurate thickness estimation.
2. sensor description
The sensor is composed of two coils, one as excitation and the other as receiver, both of which have the same dimensions, and are arranged coaxially. A schematic plot of the sensor is shown in Figure 1, with its dimensions in Table 1.
Figure 1: Sensor configuration
Table 1. Coil parameters
r1 / 11.8mmr2 / 12mm
lo (lift-off) / 0.5mm
h (height) / 3mm
g (gap) / 1mm
Number of turns N1 = N2 / 20
The connection of this sensor to an impedance analyser (SL 1260) is illustrated in Figure 2.
Figure 2: Experimental wiring schematic design
3. theoretical derivation of the compensated Peak frequency
Previously, we have proved that the peak frequency decreases with increased lift-off [30]. For completeness, main steps are summarised in this paper. It is also common knowledge that the signal amplitude also decreases with increased lift-off. Therefore, we hypothesise that an algorithm can be developed to compensate the variation in the peak frequency with the signalamplitude. In the following, we will derive such an algorithm.
We start from the Dodd and Deeds analytical solution which describes the inductance change of an air-core coil caused by a layer of non-magnetic, metallic plates [31]. Other similar formula exist [32]. The difference in the complex inductance is, wherethe coil inductance above a plate is, and is the inductance in free space.
The formulas of Dodd and Deeds are:
(1)
Where,
(2)
(3)
(4)
(5)
(6)
0 denotes the permeability of free space. N denotes the number of turns in the coil; r1 and r2 denote the inner and outer radii of the coil; while l0and h denote the lift-off and the height of the coil; and c denotes the thickness of the plate.
Equations (1-6) can be approximated based on the fact that () varies slowly with compared to the rest of the integrand, which reaches its maximum at a characteristic spatial frequency 0. The approximation is to evaluate () at 0and take it outside of the integral.
(7)
Where,
(8)
(9)
Note that in Equation (7), the sensor phase signature is solely determined by (0), which includes conductivity, the thickness of the conducting plate, and 0. ΔL0 is the overall magnitude of the signal, which is strongly dependent on the coil geometrical parameters but independent of the thickness and electromagnetic properties of the plate.
Substituting with, and considering Equation (3), Equation (8) becomes,
(10)
Assigning (11)
Equation (10) can be expressed as
(12)
In Equation (12), it can be seen that the peak frequency for the first order system is approximately and from Equation (11) it is concluded that the peak frequency increases with0.
Suppose a lift-off variation of is introduced, from Equation (6), we can see that an increase of in lift-off is equivalent to multiplying a factor:
(13)
Due to the fact that peaks at 0and that the squared Bessel term is the main contributor, a simple function with its maximum at 0is used to approximate.
(14)
wheredenotes the magnitude of the inductance change when the lift-off is zero.
This simplification is applied to obtain an analytical solution for 0.
Figure 3: Approximation of the Bessel term with a sinusoid
The shift in 0due to the effect of lift-off can be predicted as follows.
The new should maximize
and therefore
The maximum can be obtained by finding the stationary point for .
Let
And through some mathematical manipulations, a new equation can be obtained:
With small lift-off variation, holds, the right side can be approximated as .
Therefore the revised ,is
(15)
Combining (11) with (15), becomes
(16)
Combining (14) with (15), becomes
Considering andbased on small-angle approximation, is substituted with.
becomes
Substituting with
(17)
Taking natural logarithmicoperation of both sides, we arrive at:
(18)
And further:
This is now a quadratic equation with as its variable.
Therefore, the solution for is
(19)
The other solutiondoes not satisfy the small lift-off condition and therefore are discarded.
From Equ. (19), lift-off can be estimated as
(20)
Combining (16) with (20), the peak frequencywith a lift-off ofl0 becomes
(21)
Equ.(21) becomes a quadratic equation with an unknown,
And the solution is
(22)
Therefore, the original peak frequency (peak frequency priorto introducingthe lift-offl0) can be obtained by combining (11) with (22)
(23)
It can be seen in (23) that the through a compensation scheme, and using the knowledge of the peak frequency and the amplitude at a certain lift-off, the original peak frequency (peak frequency prior to introducingthe lift-offl0)can be recovered.
Further approximation can be carried out by considering. The peak frequency of the imaginary part of the inductance in Equation (21) becomes
(24)
Equation (22) becomes
(25)
Therefore, the compensatedpeak frequency (peak frequency prior to introducingthe lift-offl0) becomes
(26)
So the thickness reduces to,
(27)
Therefore from Equation (27), it can be seen that as lift-off increases, the measured peak frequency decreases, butthe numerator term also decreases to compensate for this, so that the compensated peak frequency and accurate thickness can be recovered.
4. SIMULATIONS AND EXPERIMENTS
Experiments and simulations were carried out to verify the performance of the compensation algorithm; the compensated peak frequency and the compensated thickness measurements atdifferent lift-offs were compared. Here, the imaginary part of theinductance is defined fromthe mutual impedance of the coils:
(28)
Where denotes the impedanceof the coil with the presence of ametallic plate whileis that of the coil in air.
4.1 Simulations
The simulated sensor configuration is shown in Table 1. The simulated targets are aluminium plates with a conductivity of 38.2 MS/m and thickness of 22 µm and 44 µm under varying lift-offs 0.5 mm, 2 mm, 3.5 mm and 5 mm. The simulationswere realised by a custom developed solver usingMatlab. The solver can be used to calculate the Dodd and Deeds solution (equations 1-6) and calculate the thickness and peak frequency using equations 24-27. The solver can take a range of different parameters such as frequency, sample conductivity, thickness and lift-off. In addition, the solver have been converted and packaged to an executable program.
4.2 Experimental setup
The sensor configuration and the test pieces are the same as that of simulations. And the multi-frequency response of the sensor wasobtained by a SL 1260 impedance analyser with frequency sweeping mode.
Figure 4: Simulation results as well as experimental resultsof imaginary parts of L for a 22um aluminium plate at a range of lift-offs
It can be seen from Figure 4 that the peak frequency decreases as lift-off increases. And at the same time, the magnitude of the signal decreases with increased lift-offs.
Figure 5: Simulation and experimental results:
naturallogarithmic of Im(L) ratio for the 22um sample at a range of lift-offs
(a)
(b)
Figure 6: Comparisons of as-measured (uncompensated) and compensated peak frequenciesfor the 22um aluminium plate at a range of lift-offs (a) Simulation results (b) Experimental results
As can be seen from Figure 6, the compensated peak frequency decreases following initial lift-off but remains almost constant for larger lift-offs, i.e. peak frequency is virtually immune to lift-off variations for larger lift-offs.Tests were carried out to verify this method and the thickness measurement results are shown in Table 2 for varying thicknesses and lift-offs.Here, the small thickness is defined as, so it depends on the size of the coil. In our case, this value is generally < 1mm.It can be seen from the Figure 6 that both the compensated and the uncompensated peak frequency are smaller than the actual peak frequency. Since the thickness is inversely proportional to the peak frequency (See equation (27)), therefore, the calculated thickness is larger than the actual thickness.
Table 2. Thickness measurements for different thicknesses and lift-offs
Lift-off (mm) / Actual thickness (µm) / Thickness calculated from original as measured peak frequency (µm) / Thickness calculated from Compensated peak frequency by the simple 2 coils probe (µm) / Thickness tested from previous Triple-Coil probe (µm)1.5 / 22 / 23.1 / 22.2 / 22.14
1.5 / 44 / 46.3 / 44.3 / 43.21
2 / 22 / 23.5 / 22.3 / 21.42
2 / 44 / 47 / 44.2 / 44.83
3.5 / 22 / 23.8 / 22.2 / 21.22
3.5 / 44 / 47.9 / 44.4 / 44.95
In addition, a comparison of thickness tested from previous Triple-Coil probe is also add in the table. It can be seen that the method reported in this paper has in general improved accuracy in thickness measurement and the coil has a simpler structure.
5. CONCLUSIONs
This paper has considered a compensation scheme for reducing the errors in multi-frequency eddy current measurement of metallic plate thickness. The peak frequency decreases as lift-offs increases and an algorithm has been developed, which can compensate this variation frequency and produce an index that is linked to thickness but virtually independent of lift-off.Both the simulation and experimental results show that the compensated peak frequency is almost immune to the lift-off variations. This is an important feature as lift-off variation is unavoidable in many practical applications.Although the algorithm involved is slightly more complicated,this new approach has the advantages of a less complicated mechanical configuration as well as improved accuracy, and avoids the need for a precise magnetic balance, as in the case of three-coil configuration [30].
A SL1260impedance analyser working in a swept frequency mode was used to acquire the multi-frequency data in this study. However, multi-frequency impedances can also be abstracted simultaneously using composite multi-sine waveform excitation followed by FFT operations as in [33].
Acknowledgments
The authors would like to thank the UK Engineering and Physical Sciences Research Council (EPSRC)for their financial support of this research.
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