Note: Eddy current displacement sensors independent of target conductivity Hongbo Wang ( 王洪波 ), Wei Li ( 李伟 ), and Zhihua Feng ( 冯志华 )a) Department of Precision Machinery and Precision Instrumentation, University of Science and Technology of China, Hefei, Anhui 230026, China

(Received 10 November 2014; accepted 18 January 2015; published online 30 January 2015) Eddy current sensors (ECSs) are widely used for non-contact displacement measurement. In this note, the quantitative error of an ECS caused by target conductivity was analyzed using a complex image method. The response curves (L–x) of the ECS with different targets were similar √ and could be overlapped by shifting the curves on x direction with 2δ/2. Both finite element analysis and experiments match well with the theoretical analysis, which indicates that the measured error of high precision ECSs caused by target conductivity can be completely eliminated, and the ECSs can measure different materials precisely without calibration. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4906939]

Eddy current sensors (ECSs) are widely used for noncontact measurement of vibration and displacement,1 because they are low cost, high resolution, wide bandwidth, robust, and insensitive to environmental contaminants,2,3 such as dirt, dust, oil, and water. Requirements for high performance displacement sensors increase rapidly with the development of industrial automation4,5 and micromechanical technology.6 Eddy current displacement sensors are suitable for industrial applications, but they are limited by their inhomogeneity,7 including working frequency and target selection. The working frequency of an ECS could be extremely stable, and its influence to the measurement result can be ignored. According to the working principle, the response of an ECS depends on the electric conductivity and magnetic permeability of the target;8 thus, an ECS has different measurement results for various target materials, which is called as target selectivity. The temperature variation also changes conductivity, resulting in thermal drift. Details of the thermal drift of ECSs and an effective solution have been proposed by Wang.1,9 As discussed by Wang,10 for a high-precision ECS, highconductivity, nonmagnetic metals such as aluminum or copper perform the best, so only nonmagnetic targets are discussed in present study. The calibration process of an ECS applied in industrial equipment becomes very complicated and tedious because of target selectivity, in which target conductivity may vary in every instance. Tian7 and Wang10 proposed that increasing working frequency can reduce the inhomogeneity of an ECS caused by target conductivity, but target selectivity should still be considered for most ECSs that have a working frequency between 100 kHz and 10 MHz, given that the working frequency is limited by the signal conditioning circuit and the self-resonate frequency of the sensor coil. Yu11 proposed a measuring instrument for eddy current displacement that is independent of target electromagnetic properties, which can measure the

a)Author to whom correspondence should be addressed. Electronic mail:

[email protected]. Tel.: 0086-551-63607894.

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target displacement using a coil impedance vector-projection approach immune to the target materials. However, every ECS probe has its unique impedance vector-projection angle, which is hardly obtained without tedious measurement and calculation, and more than 13% measurement error is unacceptable for a high-precision displacement sensor. Here, the quantification error of an ECS with different target materials was discussed and analyzed using the complex image method (CIM). It was found that the response curves (L–x) of the ECS with different targets were similar and√could be overlapped by shifting the curves on x direction with 2δ/2. Both finite element analysis (FEA) and experiments were conducted and the results show that this method can eliminate target selectivity problem of ECSs. In general, a typical ECS system consists of a sensing coil, a nearby conductive target, and its signal conditioning board. Figure 1(a) shows a typical relationship between the impedance of the sensor coil and the target position x of an ECS with a non-ferromagnetic target. Given that the inductance of a coil has high displacement sensitivity and better thermal stability than the resistance, inductance is usually selected as the measurand to interpret the target displacement. The responses (L–x) of an ECS to three common non-ferromagnetic target materials, namely, Cu, Al, and stainless steel (SS), are shown in Fig. 1(b). The L–x curve of the ECS with SS target is much different with the one with Al or Cu target as the conductivity of SS is only 6.7% of Cu. According to the CIM used to investigate the effect of induced eddy currents in the substrates of RF-integrated circuits,12 the target with eddy currents penetrating into it can be replaced by a superconductor image plane at a complex distance of Xcx, as shown in Fig. 2(a). The parameter Xcx can be obtained by the following equation:13 Xcx = x +

1− j tt δ · tanh[(1 + j) · ], 2 δ

(1)

where δ is the penetration depth and t t is the target thickness. For a non-magnetic target (µr = 1), the penetration depth δ relies on the working frequency f and the conductivity σ,

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Rev. Sci. Instrum. 86, 016118 (2015) TABLE I. FEA parameters and results. Conductivity

1.0%

2.5%

5.0%

10%

25%

50%

100%

δ (µm) h (µm) ∆h (µm) Errors (µm)

650 460 414 9

411 291 245 4

291 206 160 2

206 145 99 0

130 92 46 0

92 65 19 0

65 46 0 ...

FIG. 1. (a) Typical response of an ECS; (b) inhomogeneity of ECSs.

which can be expressed as δ = 1/π µ0σ f ,

(2)

where µ0 is the permeability of vacuum. Although the complex distance Xcx may not make a physical intuition, it helps in interpreting the behavior of the sensor. According to Eq. (1), the effective distance14 between the sensor coil and the target can be written as 1− j tt Xeff = x + abs{ δ · tanh(1 + j) · }. 2 δ

(3)

For displacement sensors, the target thickness should be larger than 3δ to avoid the cross-sensitivity to the target thickness, as tanh[(1 + j) · t t /δ] → 1 (t t > 3δ). As shown in Fig. 2(b), h is called as image height, which indicates the equivalent distance between the target surface and the superconductor image plane based on the CIM. Thus, the value of the image height h can be expressed as √ 1− j 2 h = abs{ δ} = δ. (4) 2 2 According to the CIM, the inductance of the ECS coil with a target only depends on the effective distance Xeff = x + h, that is, L = L(Xeff ), which means that the inductance response functions of an ECS with different targets have the following relationship: L 1(x + h1) = L 2(x + h2) = ··· = L n (x + hn ).

where ∆hi = hi − h0 is the offset value of the translation transformation. Equation (5) indicates that the response of the ECS with any new target materials Mi can be obtained as long as the response of the ECS with a standard target M0 is known. In other words, when the ECS is used to measure target Mi , the calculated displacement x could be calibrated by directly subtracting the offset ∆hi using the response of the ECS with target M0, x = f i (L) = f 0(L) − ∆hi ,

(7)

where x = f 0(L) is the function to obtain the displacement of the ECS with target M0 from the measured inductance L. This method enables ECSs to work more conveniently and easily as a displacement measuring instrument. To demonstrate the above analysis, the responses of an ECS with a target that has a conductivity from 1.0% IACS (58.0 MS/m, International Annealed Copper Standard) to

(5)

Thus, the response curves L i (x) (i = 1, 2, . . . ) of the ECS with different targets are similar, and they could be overlapped by shifting the curves on the x direction with a value of hi . Equation (5) can be rewritten as L i (x) = L 0(x + h0 − hi ) = L 0(x − ∆hi ),

(6)

FIG. 2. (a) Complex image method; (b) superconductor image plane of an ECS; (c) inductance responses of an ECS with two different targets.

FIG. 3. Response curves (L−x) of an ECS with different conductivity targets (FEA): (a) original response curves; (b) shifted response curves.

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Rev. Sci. Instrum. 86, 016118 (2015) TABLE II. Experimental parameters and results. Materials σ (MS/m) δ (µm) ∆h (µm) Errors (µm)

FIG. 4. Inductive reactance response of an ECS with different targets (experiments): (a) original response curves; (b) shifted response curves.

100% IACS have been calculated by FEA using COMSOL Multiphysics. The sensor coil of the ECS used for FEA has an inner diameter of 1 mm and an outer diameter of 5 mm, which is made up of 30-turns, 50-µm-diameter copper wire. The sensor coil has an inductance of 2.29 µH and a resistance of 1.89 Ω without target. For convenience, the inductive reactance variation ∆X = ω(L 0 − L) was used in this study to discuss the response characteristics of ECSs. According to Eqs. (2), (4), and (6), the penetration depth δ and the theoretical offset ∆h of the ECSs with different conductivity targets refer to the ECSs with 100% IACS targets were calculated, as shown in Table I. The response curves before and after shifting are shown in Figs. 3(a) and 3(b), respectively. As shown in Fig. 3(b), all seven response curves of the ECSs completely overlap. Thus, the responses of the ECS with any targets can be obtained by this offset method. The error of this method is listed in the last row of Table I, which can be ignored completely for an ECS as displacement measurement instrument, because the absolute displacement is not concerned. The measurement accuracy of the small displacement variation is important, which is determined by the sensitivity of the ECS. The sensitivity of the ECS with 1% IACS target is only 39.2% of that of the same ECS with 100% IACS target, while after the offset calibration, the sensitivity error is less than 0.1%, which can meet the requirements of the state of art displacement instrument.

Cu

Al

Brass

Ti

59.98 65 0 ...

29.23 93 21 1

10.72 154 66 2

1.33 437 277 5

A disk-shaped sensor coil with 120 turns of copper wire was manufactured and tested. The copper wire has a diameter of 50 µm. The sensor coil has an outer diameter of 6 mm and an inner diameter of 2 mm, an inductance of 47.02 µH, and a resistance of 19.72 Ω without target. The inductance response curves of the ECSs assembled in a micro-positioning stage with Ti, Brass, Al, and Cu targets were measured by a high-precision LCR meter (LCR-8101, GW Instek Co., Ltd., Taiwan), as shown in Fig. 4(a). Using Eq. (4), offset values of the ECS with Ti, Brass, and Al targets (referring to the Cu target) can be calculated out, as shown in Table II. Thus, the responses of the ECS with Ti, Brass, and Al targets are all known. As shown in Fig. 4(b), all of the inductance response curves of the ECS overlap after the shifting of the offset value, which matches quite well with the theoretical analysis and FEA results. In this paper, the measurement errors of an ECS with different target materials were discussed and analyzed using CIM. According to the CIM, the target with eddy currents penetrating into it can be replaced by a superconductor image √ plane with an offset value of 2δ/2 from the target surface. As a result, √ the responses to the image plane (with a distance of x + 2δ/2) of an ECS with different targets overlap. Thus, the response curve of an ECS with any targets can be obtained by shifting the response curve of this ECS with a standard target. The FEA and experimental results matched quite well with the theoretical analysis. Results of this study showed that an ECS can measure the displacement of any targets material precisely without calibration, which could be useful in advanced industrial applications. 1H. Wang, B. Ju, W. Li, and Z. Feng, Sens. Actuators, A 211, 98-104 (2014). 2P.

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