Measuring gaps using planar inductive sensors based on calculating mutual inductance

Sensors and Actuators A 295 (2019) 59–69

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Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna

Measuring gaps using planar inductive sensors based on calculating mutual inductance Dian Jiao a , Liwei Ni a , Xiaoliang Zhu a , Jiang Zhe a,∗ , Ziyu Zhao b , Yaguo Lyu b , Zhenxia Liu b a b

Department of Mechanical engineering, University of Akron, Akron, OH, 44325, USA School of Power and Energy, Northwestern Polytechnical University, Xi’an, 710072, China

a r t i c l e

i n f o

Article history: Received 12 February 2019 Received in revised form 26 April 2019 Accepted 16 May 2019 Available online 21 May 2019 Keywords: Inductive sensor Gap measurement Mutual inductance Planar coil Eddy current Calibration

a b s t r a c t This paper presents a new method for planar inductive sensor to measure the gap between the sensor and a non-ferrite metallic target. The eddy current on the target plate is modeled as a virtual coil; the inductance change of the planar sensing coil is a result of the mutual inductance of the sensing coils and the virtual coil. We introduce a method to calculate the gap from measured coil inductance by studying the mutual inductance between the sensing coil and the virtual coil. From our analysis, we found that with this method only one calibration curve is needed for measuring the gap between the sensing coil and a metallic target plate made of different materials. When the target material is changed, the new calibration curve can be obtained by adding a constant to the base calibration curve. In comparison, traditional planar inductive sensors require a family of calibration curves for measuring gaps from targets made of different materials. To verify the validity of the method, three planar proximity sensors with different dimensions were manufactured and used to measure the gap from four different non-ferrite targets, titanium, copper, zinc and aluminum. Results showed that in a large measurement range (500 ␮m–5000 ␮m), the calibration was simplified; the calculated gaps using the method and the actual gaps were in good agreement, with a maximum error of 3.2%. The method is valid regardless of the dimensions of planar coils. It can be used for facilitating measuring gap and detecting metallic objects in machines and automation equipment made of various materials. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Planar coil inductive sensors, also called 2D eddy-current sensors, have been widely used for gap/displacement measurement [1]. Compared to other types of sensors, Inductive sensors have obvious advantages including low cost, easy installation, and noncontact measurement [2]. In addition inductive sensors provides good accuracy and stabilization under high temperature, high pressure and filthy conditions [3]. For example, the eddy-current sensors can be used to measure the tip clearance of the turbine blade at a high temperature environment [4]. To measure the gap or displacement between a sensing coil and a target metal plate, an AC signal is applied to the sensing coil and a magnetic field is generated. When a metal plate is present near the sensing coil, an eddy current is induced in the metal plate, which generates a magnetic field opposite to that generated by the sensing coil. In turn it causes an inductance decrease of the sensing coil,

∗ Corresponding author. E-mail address: [email protected] (J. Zhe). https://doi.org/10.1016/j.sna.2019.05.025 0924-4247/© 2019 Elsevier B.V. All rights reserved.

which can be measured in terms of output voltage. The smaller the gap, the larger the decrease in the inductance. Traditional inductive sensors rely on calibrating the relations between the coil inductance and the gap [5]; from the calibration curves, unknown gaps can be back calculated from the measured coil inductance without calculating the complex mutual inductance between the sensing coil and a target plate. However, there are two limitations of the traditional calibration methods. First, if a target material is changed, the calibration curve needs to be rebuilt because the eddy current varies in a different material [5]. This is also true for measuring the gap from a metallic target at various temperatures [6]. Hence, the tedious calibrations have to be conducted to obtain a family of calibration curves for each material or at each measured temperature. Second, because of the complex relationship between the inductance and the gap, to provide good accuracy and simplify the calculations, the calibration curves are typically divided into discrete sections; each section covers a small gap range which can be approximated as a straight line. While using separate sections of calibration curves can provide accurate measurements [5], calibration needs to be done at many gaps with fine incremental step, making the calibration very time consuming. In addition, if one

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D. Jiao et al. / Sensors and Actuators A 295 (2019) 59–69

Nomenclature Intermediate variable in Eq. (11) Gap between the sensing coil and the target surface Complete elliptic integral of the first kind Excitation frequency Intermediate variable in Eq. (4) Depth of penetration Intermediate variable in Eq. (12) Complete elliptic integral of the second kind Equivalent inductance of the sensing coil Base inductance of the sensing coil without any  r4  influence from the eddy current Intermediate variable in Eq. (4) L r 3 Lt Equivalent inductance of the virtual coil. li (i = 1,2,3,4) The intermediate variables in Equations (12–21), l1 = l4 = r3, l2 = l3 = r4 M Mutual inductance between the sensing coil and the virtual coil N1 Number of turns of the sensing coil Number of turns of the virtual coil N2 R Equivalent resistance of the sensing coil Base resistance of the sensing coil without any influRc ence from the eddy current Equivalent resistance of the virtual coil Rt r1 Inner radius of the sensing coil r2 Outer radius of the sensing coil r3 Inner radius of the virtual coil Outer radius of the virtual coil r4 t Thickness of the target plate Intermediate variable in Eq. (4) a ˇ Intermediate variable in Eq. (3) Intermediate variable in Eq. (12) i,n i (i = 1,2,3,4) The intermediate variables in Eqs. (12)–(21), 1 = 2 = r1, 3 = 4 = r2  Electrical conductivity of the target material  Heuman’s Lambda function in Eq. (12) An d E f   r H r4 3 h Jin K L Lc

does not know the exact range of the gap to be measured, choosing an incorrect section of the calibration curve may cause large measurement error. To overcome the above calibration problems, efforts have been made to evaluate the gap based upon studying the mutual inductance between the sensing coil and the target plate. Several new calibration methods or algorithms were attempted by different researchers [6–8]. Because the calculation of the mutual inductance is complex, containing too many unknowns, to date most of these methods were used to qualitatively evaluate the influence of one parameter/condition on the gap measurement, but cannot be used to calculate the gap from the inductance change mathematically [1]. Recently, finite element analysis (FEA) was used to analyze the equivalent resistance and inductance of the virtual coil and distribution of the eddy current with a known gap [7]. However, it is impractical to use the FEA to back calculate the unknown gap from the measured coil inductance for gap measurement applications. In this paper, based on prior efforts [7,9], we first introduce a mathematical model to model the mutual inductance between a sensing coil and a target plate. We then simplify the model and propose a new calibration method based on the model. Compared to the traditional calibration methods, the new calibration method allows us to obtain the calibration curve for a different material in a single step without recalibrating the curve. Additionally it does not

need to be divided the calibration curve into many sections while still providing decent accuracy. 2. Mathematical model and calculation method Fig. 1(a) illustrates the working principle of the inductive planar coil sensor for gap/displacement measurement from a metallic plate. The sensing coil has an inner and outer radii of r1 and r2 . When the sensing coil is excited by a high frequency AC signal, a magnetic field is generated, which induced an eddy current in the metal plate. Prior work by Vyroubal [9] indicated that eddy current in a metal plate can be regarded as a planar coil. Thus, here we modeled the eddy current as a virtual coil with inner and outer radiuses of r3 and r4 [9,10]. With a fixed excitation frequency, if the gap (d) between the sensing coil and the target is varied, the equivalent inductance and resistance of the sensing coil is changed due to mutual induction change between the coil and the target metal plate. The gap measurement can be considered as the mutual inductance problem of the two coils: the sensing coil and the virtual coil (Shown in Fig. 1b). Assuming the planar sensing coil and the target are parallel (Shown in Fig. 1b), the directions of two magnetic fields are opposite. Thus the magnetic flux of the sensing coil is reduced at the presence of the eddy current. The smaller the distance, the larger the reduction in the magnetic flux of the sensing coil, which is reflected as the change of the equivalent inductance of the sensing coil. The equivalent inductance (L) and the equivalent resistance R of sensing coil is derived from Kirchhoff’s Law [12–14]: L = LC − R = RC +

ω2 M 2 Lt

(1)

Rt2 + ω2 Lt2 ω2 M 2 Rt

(2)

Rt2 + ω2 Lt2

where Lc and Rc are the values of the inductance and resistance of the sensing coil without any influence from the eddy current, M is the mutual inductance between the sensing coil and the virtual coil, Lt and Rt are the self-relative inductance and resistance of the target generated by the eddy current (or equivalent inductance and resistance of the virtual coil, ω is excitation frequency in rad/s (ω = 2  f ). The expansion formula of the equivalent inductance of the virtual coil can be described by Eq. (3), originally derived by Slobodan Babic et al [15]: Lt =

0 N2 2 r 3( r4 3

r

− 1)( r4 − 1)

·ˇ

(3)

3

Where ˇ is a function of r3 and r4 :

 r 3

ˇ=[

4

r3

 r 3

+4

4

r3

+ 1] [0.832 − E (a)] −

+4

 r 4 4

+[(−

−

2

a =

r3

r  4

2

4

3

+ 1]a2 − 4E (a) ·

 r 3

−2

 r 3   4 r4 r3 H

r3

r3

4

r3 −L

r 

4 + 1  r4 4  r log (2) + 3r  {[ 2 r3 2 r4 a2

−2

r  4

r3

r  4

r3

 r   r 2 4 4 r3

[

r3

 r 3

− 1)a2 + 4

4

r3

+ 1]

+ 4]K (a)}

(4)

r 

[1 +

4

r3

r  4

r3

]

2

(5)

D. Jiao et al. / Sensors and Actuators A 295 (2019) 59–69

61

Fig. 1. (a) Working principle of gap measurement. Eddy current is generated in the target metal plate and cause an inductance change of the sensing coil. (b) The eddy current effect can be modeled as a virtual planar coil on the target plate. (c) Equivalent circuit model of the measurement system. (d) Mathematical model in Neumann’s formula [11] to calculate mutual inductance between the two coils.

H

r  4

r3





 2

=

1+ log  1+

0

 r 2 4

r3

 r 2 4 r3

+2

r 

−2

4

r3

r  4 r3

cosx + 1 + sinx +

r 

r  4 r3

4

r3

L

r4 r3



 2

=

log 0

 2 1+

r4 r3

  +2

r4 r3

dx

sinx − 1



  cos2x +

r4 r3

2 r

hln r4

+ cos2x

dx

1

(11)

(7) Where:

(8)

Where h is the depth of the penetration which can be expressed by [16]:



(10)

A1 − A2 + A3 − A4 (0 N1 N2 ) (3r2 − 3r1 ) (r4 − r3 )

2ln n An = − 

(9)



ln n

4ln n

·E

2

(ln + n ) + d2

4ln n

+ln n



4 2

(ln + n ) + d2

2

(ln + n ) +



·

d2 − ln2 − n2 + n

4ln n

·K

f

To solve gap d from Eq. (1), based on the Neumann’s formula, the basic formula for calculating the mutual inductance of two planar coils can be expressed as [17–19]:



(ln +n )2 +d2

3

h≈

ds1 · → ds2 d

Where d s1 and d s2 are incremental sections of the filaments (shown in Fig. 1d). The integration of Eq. (10) can be given by expression (11), originally derived by Slobodan Babic et al [19]: M=

E and K are the complete elliptic integral of the first and second kind. r3 and r4 are the inner radius and outer radius of the virtual coil. N2 is the number of turns of the virtual coil. The formula to determine the equivalent resistance of the virtual coil is expressed as [7], originally derived by Wang et al: Rt =



cosx

(6)

  

0 4

M=



1,n,





4ln n 2

4ln n 2

(ln + n ) +

d2

 ln2 + d2

2



2,n,



n2 + d2

− signum

(ln + n ) + d2

n2 + d2 + ln





1−

1−

d2

  n

−  d 





 n2 + d2 − ln

  ln

−  d 



ln2 + d2 2

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D. Jiao et al. / Sensors and Actuators A 295 (2019) 59–69



3,n,





·



1−

1−

4ln n 2

(ln + n ) +

− signum

d2





4ln n

4,n,

2

(ln + n ) + d2





Table 1 The dimensions and parameters of three sensing coils used in the experiments.

ln2 + d2 − n

d3 J1n + n3 J2n + ln3 J3n n − 2

(12)

= 1, 2, 3, 4.





 2

J1n =

 2

f (ˇ)dˇ = 0

0

⎡ 1 sin(ˇ)

 

d2 cos ˇ − ln n sin

 2

J2n =

 2

J3n =

ˇ

⎛

ln2 + n2 + d2 − 2ln n cos ˇ

 

⎦ dˇ n = 1, 2, 3, 4.

arcsinh ⎝ 

 ⎞ ⎠ dˇn = 1, 2, 3, 4. (14) 



n2 sin2 2ˇ + d2

⎛

 ⎞ ⎠ dˇn = 1, 2, 3, 4. (15) 



n + ln cos 2ˇ

arcsinh ⎝ 



ln2 sin2 2ˇ + d2

 f (0) =

ln2 + n2 + d2 + 2ln n −



ln2 + n2 + d2 − 2ln n

d

n

= 1, 2, 3, 4.

 2

(13)

= 2arctan



(16)

d



1,n = arcsin



ln n ln2

+ n2 + d2

n = 1, 2, 3, 4.

(17)

, n = 1, 2, 3, 4.

(18)



 d |

n2 + d2 + n

⎛ ⎞  4n2 − + 1 2  ⎜ ⎟ ⎜ ⎟ n2 +d2 +n  ⎜ ⎟ 2,n = arcsin ⎜ ⎟ , n = 1, 2, 3, 4. ⎜ − 4ln 2n 2 + 1 ⎟ ⎝ ⎠ (ln +n ) +d

(19)

Coil 1

Coil 2

Coil 3

1.4 mm 12.92 mm 8 1V 1 MHz 500-5000 ␮m

1.4 mm 9.02 mm 6 1V 1 MHz 500-5000 ␮m

1.4 mm 14.20 mm 10 1V 1 MHz 500-5000 ␮m

Thus M is a function of d, r1 , r2 , r3 , r4 , N1 and N2 . Eq. (1) can be written as follows: 2

L = LC −



0

f

 

ln + n cos 2ˇ

0



ˇ

ln2 + n2 + d2 + 2ln n cos ˇ

dsin(ˇ)



 

⎤

  2



2



dsin(ˇ)

 

−arctan

d2 cos ˇ + ln n sin

⎣arctan

Inner diameter Outer diameter Number of turns Excitation voltage Excitation frequency Gap range

ω2 M(d, r 1 , r2 , r3 , r4 , N1 , N2 ) Lt (r3 , r4 , N2 ) 2

Rt (r3 , r4 ) + ω2 Lt (r3 , r4 , N2 )2

(26)

The unknowns are r3 and r4 (the inner and outer radius of the virtual coil), d (the gap between the center of the two coils), and N2 (the number of turns of the virtual coil). Next, we present a method to solve d from Eq. (26) with a measured L. At a fixed gap between the sensing coil and the virtual coil, the eddy current generated on the target plate (and thus the Lt and Rt ) is determined by r3 , r4 and N2 . From Eq. (3), the equivalent inductance of the virtual coil (Lt ) is determined by N2 (the coil turns of the virtual coil), r3 and r4 . The Lt change caused by the variations in r3 and r4 can be achieved by changing N2 . In addition, prior finite element study [7] indicated that when the gap is small, the change in the inner and outer radius of the virtual coil (r3 and r4 ) is less than 10% when the gap d varies, implying that r3 and r4 can be assumed to be nearly constant when the gap changes. From the above, we assume that r1 is equal to r3 , and r2 is equal to r4 . If the gap changes, the Lt variation can be attributed to the change of N2 . The wire radius of the virtual coil can be considered as a variable, such that r3 and r4 can remain the same when the coil turns of the virtual coil varies. With this assumption, in Eq. (26) there are only two unknowns left, the gap d and the number of turns of the virtual coil (N2 ). By taking a number of measurements of equivalent inductances (L) of the sensing coil at a number of gaps (say 6–7 gaps), we can build a calibration curve between d and N2 . After the d - N2 relationship curve is built, an unknown gap can be calculated from a measured coil inductance by coupling this relationship with Eq. (26). 3. Experimental setup, measurement and calculation results 3.1. Experimental setup

 3,n = arcsin





 d | ln2

+ d2

+ ln

, n = 1, 2, 3, 4.

⎛ ⎞  4ln2 − + 1  2  ⎜ ⎟ ⎜ ⎟ ln2 +d2 +n ⎜ ⎟ 4,n = arcsin ⎜ ⎟ , n = 1, 2, 3, 4. ⎜ − 4ln 2n 2 + 1 ⎟ ⎝ ⎠ (ln +n ) +d

(20)

(21)

r1 = 1 = 2

(22)

r2 = 3 = 4

(23)

r3 = l1 = l4

(24)

r4 = l2 = l3

(25)

 is Heuman’s Lambda function.

Experiments were conducted to validate the proposed calculation method. The experimental setup is showed in Fig. 2. A sensing coil made of 0.45mm-in-diameter copper wires attached on a precision X–Y-Z stage (90 × 90 × 90 mm manual precision linear stage) was positioned above a metal plate. The metal plate is 5 cm × 5 cm × 5 cm. The X–Y-Z stage can regulate the gap between the sensing coil and the metal plate with a 10 ␮m resolution. The inductance change of the sensing coil was induced by the gap change. We chose three different coils to prove that the proposed calculation method is valid for various planar coils regardless of their sizes. The dimensions and parameters of the sensing coils are listed in Table 1. As shown in Fig. 2, each coil has a 500 ␮m thick ceramic layer (made of ceramic 645-N) on top to protect the coil and preserve its shape (such that the base inductance of each coil remains unchanged in all experiments). Four metal plates made of titanium, zinc, copper and aluminum were used to study the influence of the material change. Their conductivities cover the conductivity

D. Jiao et al. / Sensors and Actuators A 295 (2019) 59–69

63

Fig. 2. Illustration of the experiment setup and a picture of one coil made of copper wire. A 500 ␮m thick ceramic layer was applied on top of the coil surface to protect the sensing coil and keep the shape of the coil so that the coil’s base inductance remains unchanged in all measurements.

Fig. 3. Measured inductance of sensing coil 1 as a function of the gap d between the sensing coil and the target metal plate.

Fig. 4. Calculated number of turns of the virtual coil (N2 ) as a function of the gap d.

3.2. Measurement and calculation results

range of most commonly used metallic materials. In all experiments, an LCR meter (Keysight Technologies E4980A1-030) was used to record the equivalent inductance of the planar coils at various gaps. The excitation signal had a frequency of1MHz and a 1 V peak-to-peak magnitude. To prove the electrical equivalent circuit model (R in series with L, see Fig. 1(b)) is valid for the sensing coils, we measured their phase angles and impedance using a precision LCR meter. At three excitation frequencies, 0.5 MHz, 1 MHz and 2 MHz, the phase angles of the sensing coils varied from 87.2 to 88.0 degrees, showing the coil inductance was dominant. In addition, the impedance of the sensing coil was nearly doubled when the excitation frequency was increased from 0.5 MHz to 1 MHz and from 1 MHz to 2 MHz, indicating the impedance of the coil dominantly came from the coil inductance (2fL). Hence the serial R-L model can be used at the working frequencies (0.5 MHz to 2 MHz).

The experiments were conducted as the following: First, we measured the base inductance of the sensing coil (Lc ) without presence of any metallic object. Next, we measured the inductance of the coil (L) when a 5 cm × 5 cm × 5 cm titanium plate was placed below the sensing coil. Using the precision X–Y stage, the gap between the sensing coil and the target was regulated from 500 ␮m to 5000 ␮m. Fig. 3 shows the measured inductance of the sensing coil as a function of the gap between the sensing coil and the metal plate. The inductance decreases as the gap decreases because of the increased mutual inductance. Applying all knowns (Rt , r1 , r2 , r3 , r4 , N1 , etc) in Eq. (26), with the measured coil inductance for each known gap d, N2 was solved by using bisection iteration from Eq. (26). With a set of measured inductances at known gaps, we built a relationship curve between the N2 and d. For metal plates made of different materials (zinc, copper and aluminum), N2 - d relationship were obtained by repeating the same process. The results of the N2 -d curves are shown in Fig. 4. Each curve was built by 8–9 data

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D. Jiao et al. / Sensors and Actuators A 295 (2019) 59–69

Fig. 5. (a) Calculated N1 /N2 as a function of the gap (d) between the sensing coil and the target metal plate. The N1 /N2 - d curve shifts in parallel with a parallelism distance C when target material is changed. (b) The constant C for zinc, aluminum and copper targets.

Fig. 6. (a) Calculated N1 /N2 as a function of d for sensing coils 2. (b) The constant C for zinc, aluminum and copper targets.

points. Fig. 4 shows that as the gap increases, N2 decreases due to decreased eddy current. Similarly, at the same gap, N2 is higher for a material with higher conductivity. This is because eddy current/mutual inductance is increased with increased conductivity [13,20,21]. While Fig. 4 shows each non-ferrite material has its own N2 - d curve, next we converted the vertical axis from N2 to N1 /N2 , where N1 is the number of turns of the sensing coil (N1 = 8 for sensing coil

1). Fig. 5 show the curves of N1 /N2 vs d for different materials. We found that the N1 /N2 - d curve moves in parallel when the material is changed. In other words, there is a parallelism distance C of N1 /N2 regardless of the gap d when the material of the target plate is changed:

C=(

N1 N1 ) −( ) N2 m2 N2 m1

(27)

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65

Fig. 7. (a) Calculated N1 /N2 as a function of d for sensing coil 3. Figure 5, 6 and 7 showing that N1 /N2 – d curves for different materials are in parallel. There is a constant parallelism distance between the two curves for different materials. (b) The constant C for zinc, aluminum and copper targets.

Fig. 8. Comparisons of the calculated gaps (from Eq. (27)) to the actual gaps. (a) Gap between sensing coil 1 and the zinc target; N1 /N2 - d relation was obtained from N1 /N2 - d relation of titanium by adding a constant -0.0669. (b) Gap between sensing coil 1 and the aluminum target; N1 /N2 - d relation was obtained from N1 /N2 - d relation of titanium by adding a constant -0.1009. (c) Gap between sensing coil 1 and the copper target; N1 /N2 - d relation was obtained from N1 /N2 - d relation of titanium by adding a constant -0.1333.

where subscription m1 and m2 represent material 1 and material 2. Using titanium as the base material (material 1), we calculated the C at different gaps. Fig. 6 and 7 also shows the results indicating the C is a constant with small variations for different geometry of sensing coils. Where this is not mathematically proved due to complex

relation between N1 /N2 and d, this finding significantly simplify the calibration process for different materials: only one calibration curve for a base material is needed; the calibration curves for other materials can be determined by adding a constant C to the base curve.

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D. Jiao et al. / Sensors and Actuators A 295 (2019) 59–69

Fig. 9. Comparisons of the calculated gaps to the actual gaps. (a) Gap between sensing coil 2 and the zinc target; N1 /N2 - d relation was obtained from N1 /N2 - d relation of titanium by adding a constant -0.0413. (b) Gap between sensing coil 2 and the aluminum target; N1 /N2 - d relation was obtained from N1 /N2 - d relation of titanium by adding a constant -0.0601. (c) Gap between sensing coil 2 and the copper target; N1 /N2 - d relation was obtained from N1 /N2 - d relation of titanium by adding a constant -0.0749.

To prove if the finding is valid for sensing coils with different geometries, same measurements were conducted using the sensing coils 2 and 3. Their dimensions are listed in Table 1. Four materials, titanium, zinc, aluminum and copper were used. Their conductivities cover the conductivity range of most commonly used metallic materials. The results are shown in Figures 5, 6 and 7, indicating that even the geometries of the sensing coils are quite different, the finding that the N1 /N2 - d curves are parallel for different materials seems still valid. For coil 2, the C is -0.0413 ± 0.0016 for zinc, -0.0601 ± 0.0038 for aluminum and -0.0749 ± 0.0009 for copper target. For coil 3, the C is -0.0729 ± 0.0057 for zinc, -0.1096 ± 0.0047 for aluminum and -0.1443 ± 0.0082 for copper target. When one calibration curve for one material and the constant C is determined, the calibration curves for a different material can be obtained. Note that the parallelism distance C can be determined at one gap, avoiding generating a complete new curve with many calibration data points. Next to validate this method, we will evaluate the accuracy of the gap measurements using the obtained N1 /N2 - d curves.

×10−14 d4 − 2.707 × 10−11 d3 + 2.911 ×10−8 d2 − 1.303 × 10−5 d + 1.3262

(

N1 N1 ) =( ) − 0.1333 N2 Cu N2 Ti

(29)

(

N1 N1 ) =( ) − 0.1009 N2 Al N2 Ti

(30)

(

N1 N1 ) =( ) − 0.0669 N2 Zn N2 Ti

(31)

For Coil 2 (9.02 mm in diameter): (

(

N1 ) = 1.631 × 10−22 d6 − 2.588 × 10−18 d5 + 1.507 N2 Ti

N1 ) = 1.156 × 10−21 x6 − 1.4028 × 10−17 x5 + 5.931 N2 Ti ×10−14 x4 − 0.754 × 10−10 x3 + 0.466 × 10−8 x2 −0.0000372x + 0.919

(32)

(

N1 N1 ) =( ) − 0.0749 N2 Cu N2 Ti

(33)

(

N1 N1 ) =( ) − 0.0601 N2 Al N2 Ti

(34)

(

N1 N1 ) =( ) − 0.0413 N2 Zn N2 Ti

(35)

4. Validation of the gap measurements with the N1 /N2 - d curves All (N1 /N2 - d curves) (for the three sensing coils and different materials) are correlated to 6th -order polynomials as shown below: For Coil 1 (12.9 mm in diameter):

(28)

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67

Fig. 10. Comparisons of the calculated gaps to the actual gaps. (a) Gap between sensing coil 3 and the zinc target; N1 /N2 - d relation was obtained from N1 /N2 - d relation of titanium by adding a constant -0.0729. (b) Gap between sensing coil 3 and the aluminum target; N1 /N2 - d relation was obtained from N1 /N2 - d relation of titanium by adding a constant -0.1096. (c) Gap between sensing coil 3 and the copper target; N1 /N2 - d relation was obtained from N1 /N2 - d relation of titanium by adding a constant -0.1443.

For Coil 3 (14.2 mm in diameter): N1 ( ) = 8.862 × 10−23 x6 − 1.836 × 10−18 x5 + 1.441 N2 Ti ×10−14 x4 − 3.901 × 10−11 x3 + 0.776 ×10−7 x2 − 0.867 × 10−5 x + 1.951

(36)

(

N1 N1 ) =( ) − 0.1443 N2 Cu N2 Ti

(37)

(

N1 N1 ) =( ) − 0.1096 N2 Al N2 Ti

(38)

(

N1 N1 ) =( ) − 0.0729 N2 Zn N2 Ti

(39)

Note that with one sensing coil, when a material is changed, the N1 /N2 - d curve shift up or down in parallel with a parallelism distance C. Because C is a constant, it can be determined only at one gap; this significantly simplifies the calibration process. Next, we prove the N1 /N2 - d curves can be used to calculate the gap accurately from the measured coil inductance. The procedures were: we 1) took measurements of the inductance of the sensing coil at many positions from 500 ␮m to 5000 ␮m, 2) applied the inductance L, the base inductance Lc , all known parameters (r1 , r2 , r3 , r4 and N1 ) and coupled the relationship the N1 /N2 - d with Eqs. (26), and (3) used Bi-section method to calculate the gap d. The comparisons between the calculated gaps and the actual gaps for various materials and sensing coils are given in Fig. 8.

Fig. 8(a) shows the gap comparison between sensing coil 1 and the zinc target; N1 /N2 - d relation was obtained from N1 /N2 - d relation of Titanium by adding a constant -0.0669. The calculated gaps are in good agreement with the actual gaps; the maximum error, 2.50%, occurred at an actual gap of 840 ␮m for zinc target (Fig. 8(a)). When the target material was changed to aluminum, using the proposed calculation method, the maximum error was 2.43% at an actual gap of 4040 ␮m (Fig. 8(b)). When the target material was changed to copper, the maximum error was 3.19% at an actual gap of 690 ␮m (Fig. 8(c)). Fig. 9(a), 9(b) and 9(c) show the calculated gap compared to the actual gap for coil 2. For coil 2, the maximum error was 2.69% occurring at an actual gap of 520 ␮m displacement for zinc target (Fig. 9(a)). A maximum error 3.07% for aluminum target occurred at an actual gap of 1240 ␮m (Fig. 9(b)). When the target material was changed to copper, the maximum error was 2.45% at an actual gap of 940 ␮m (Fig. 9(c)). The maximum error for coil 3 was 2.60% occurring at 4040 microns displacement for zinc target (Fig. 10(a)). As the target material was changed to aluminum, using the proposed calculation method, the maximum error was 2.53% at an actual gap of 3240 microns (Fig. 10(b)). A maximum error 2.49% occurred at an actual gap of 4440 ␮m for a copper target (Fig. 10(c)). All results show that the calculated gaps are in good agreement with the actual gaps for different materials and coils. This indicated that although there exist small variations on C, when C is considered as a constant to back calculate the gap d, the prediction in d matches well with the actual d. Thus C can be taken as a constant to simplify the calibration without sacrificing the measurement accuracy.

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5. Discussions The result proves the feasibility of the method/algorithm presented in previous section to calculate the gap with simplified calibration. The error can be further reduced by 1) adding a few more data points in the L vs d measurements, and 2) increasing the times of iteration while calculating N2 from L in Eq. (26) to obtain a more N1 /N2 - d base curve, and calculating d by coupling N1 /N2 - d curve with Eq. (26). The results also indicate that the parallelism distance C is dependent on both the target material and the geometry of the coil. From the data showed in Figures 5, 6 and 7, coil 3 with the largest outer diameter has the largest value of C for the same target material. In addition, it seems that the smaller the conductivity, the larger the value of C for the same coil. For a coil with fixed geometry, only one base calibration curve for one material is needed. When the coil is used for measuring the gap from a different target material, we only need to take one measurement at one known gap to obtain C for this material, instead of obtaining a new calibration curve by taking many measurements at different gaps. This significantly simplifies the calibration. Only if the sensing coil is changed, a new calibration curve is needed for this coil. Compared to the traditional method which relies on calibrating an inductance-gap curve for each material, the calibration process is significantly simplified and can back calculate the gap using only one base calibration curve for various materials. Worth mentioning here that the method is based on an assumption that the thickness of the target metal plater is larger than the penetration depth, h. From Eq. (9), the penetration depth is affected by the excitation frequency. When the measured target is thin, the excitation frequency can be adjusted to satisfy the assumption, i.e. target thickness > penetration depth (t > h). 6. Conclusions This paper proposed a new method to calculate the gap between the non-ferrite materials and planar proximity sensor. The method is based on studying the mutual induction between a planar sensing coil and the eddy current on a metal plate; the latter can be modeled as virtual coil. We found that the N1 /N2 -d curve for different materials are in parallel with a constant parallelism distance, where N1 and N2 are the coil turns of the sensing coil and the virtual coil. Hence the N1 /N2 - d curve for a new material can be obtained by adding a constant C to an existing curve, where C can be calibrated at only one gap. Experiments indicated that this finding is valid regardless of the dimension of the sensing coil. With this method, 2D planar proximity sensor can be used to measure gaps with a larger measurement range (500 ␮m–5000 ␮m). Our gap measurement results showed with this range, the measurement error of gap is well within 3.2%. This method significantly simplify the calibration process when target material is changed. We expect it can be used for facilitating measuring gap and detecting metallic objects in machines and automation equipment made of various materials. Acknowledgments D. Jiao, L. Ni and J. Zhe acknowledge the partial support from National Science Foundation of USA under grants ECCS1625544 and ECCS1905786. References [1] S. Tumanski, Induction coil sensors - a review, Meas. Sci. Technol. (2007), http://dx.doi.org/10.1088/0957-0233/18/3/R01. [2] Z. Xiao, W. Hu, C. Liu, H. Yu, C. Li, Noncontact Human-Machine Interface With Planar Probing Coils in a Differential Sensing Architecture, 2018, http://dx.doi. org/10.1109/TIM.2017.2784079.

[3] J. Poliakine, Y. Civet, Y. Perriard, Design and manufacturing of high inductance planar coils for small scale sensing applications, Procedia Eng. (2016), http:// dx.doi.org/10.1016/j.proeng.2016.11.368. [4] L. Du, X. Zhu, J. Zhe, A high sensitivity inductive sensor for blade tip clearance measurement, Smart Mater. Struct. 23 (2014), http://dx.doi.org/10.1088/ 0964-1726/23/6/065018. [5] S. Zuk, A. Pietrikova, I. Vehec, Development of planar inductive sensor for proximity sensing based on LTCC, Proc. Int. Spring Semin. Electron. Technol. (2016), http://dx.doi.org/10.1109/ISSE.2016.7563231. [6] Y. Han, X. Zhu, C. Zhong, J. Zhe, Online monitoring of dynamic tip clearance of turbine blades in high temperature environments, Meas. Sci. Technol. 29 (2018). [7] H. Wang, W. Li, Z. Feng, Noncontact thickness measurement of metal films using eddy-current sensors immune to distance variation, IEEE Trans. Instrum. Meas. (2015), http://dx.doi.org/10.1109/TIM.2015.2406053. [8] H. Wang, Y. Liu, W. Li, Z. Feng, Design of ultrastable and high resolution eddy current displacement sensor system, in: 40th Annu. Conf. IEEE, 2014. [9] D. Vyroubal, Impedance of the eddy-current displacement probe: the transformer model, IEEE Trans. Instrum. Meas. (2004), http://dx.doi.org/10. 1109/TIM.2003.822705. [10] C.V. Dodd, W.E. Deeds, Analytical solutions to eddy-current probe-coil problems, J. Appl. Phys. (1968), http://dx.doi.org/10.1063/1.1656680. [11] M. Chaoui, H. Ghariani, M. Lahiani, F. Sellami, Maximum of mutual inductance by inductive link, Proc. Int. Conf. Microelectron. ICM (2002), http://dx.doi.org/ 10.1109/ICM-02.2002.1161544. [12] G.Y. Tian, Z.X. Zhao, R.W. Baines, The research of inhomogeneity in eddy current sensors, Sens. Actuators, A Phys. (1998), http://dx.doi.org/10.1016/ S0924-4247(98)00085-5. [13] H. Wang, B. Ju, W. Li, Z. Feng, Ultrastable eddy current displacement sensor working in harsh temperature environments with comprehensive self-temperature compensation, Sens. Actuators, A Phys. (2014), http://dx.doi. org/10.1016/j.sna.2014.03.008. [14] J. García-Martín, J. Gómez-Gil, E. Vázquez-Sánchez, Non-destructive techniques based on eddy current testing, Sensors (2011), http://dx.doi.org/ 10.3390/s110302525. [15] S. Babic, C. Akyel, Improvement in calculation of the self- and mutual inductance of thin-wall solenoids and disk coils, IEEE Trans. Magn. (2000), http://dx.doi.org/10.1109/TMAG.2000.875240. [16] C.R. Neagu, H.V. Jansen, A. Smith, J.G.E. Gardeniers, M.C. Elwenspoek, Characterization of a planar microcoil for implantable microsystems, Sens. Actuators A Phys. (1997), http://dx.doi.org/10.1016/S0924-4247(97)01601-4. [17] C. Akyel, S.I. Babic, M.-M. Mahmoudi, Mutual inductance calculation for non-coaxial circular air coils with parallel axes, Prog. Electromagn. Res. (2009) 287–301, http://dx.doi.org/10.2528/PIER09021907. [18] C. Akyel, S. Babic, Mutual inductance between coaxial circular coils of rectangular cross section and thin coaxial circular coils with constant current density in air (filament method), in: Proc. 6th WSEAS Int. Conf. Appl. Electr. Eng., 2007. [19] S. Babic, S. Salon, C. Akyel, The mutual inductance of two thin coaxial disk coils in air, IEEE Trans. Magn. (2004), http://dx.doi.org/10.1109/TMAG.2004. 824810. [20] R.J. Ditchburn, S.K. Burke, Planar rectangular spiral coils in eddy-current non-destructive inspection, NDT E Int. (2005), http://dx.doi.org/10.1016/j. ndteint.2005.04.001. [21] Y.P. Su, X. Liu, S.Y.R. Hui, Mutual inductance calculation of movable planar coils on parallel surfaces, IEEE Trans. Power Electron. (2009), http://dx.doi. org/10.1109/TPEL.2008.2009757.

Biographies Dian Jiao received his BS and MS in Mechanical Engineering from Central Connecticut State University and University of Delaware in 2015 and 2017, respectively. He is currently a Ph.D. candidate in Mechanical Engineering at the University of Akron. His research interests include sensors for dynamic gap measurements as well as for machine health monitoring. Liwei Ni received his BS and MS degree in Aerospace Engineering from Nanjing University of Aeronautics and Astronautics and the Northwest Polytechnical University, China, in 2013 and 2016 respectively. He is currently working toward a Ph.D. degree in Mechanical Engineering at the University of Akron. His major research focuses are microfluidic devices for cell analysis and sensors for dynamic gap measurement. Xiaoliang Zhu received his MS in Electrical Engineering from Beijing University of Technology in 2012, and Ph.D. degree in Mechanical Engineering from University of Akron in 2016 with a focus on integrated sensing systems for machine health monitoring. He is currently a R&D Researcher in Hitachi America Ltd working on automatic systems using machine vision and image processing for quality inspection of automotive products. Jiang Zhe is a Professor of Mechanical Engineering at The University of Akron, Ohio. He received his Ph.D. Degree in Mechanical Engineering from Columbia University in 2002. Prior to joining University of Akron, he was a research and development engineer in Fitel Technologies in 2002 and Advanced Microsensors in 2003. His major research areas are micro and nano sensors, microfluidic devices and lab-ona-chip devices. He was elected to the Fellow of ASME in 2014.

D. Jiao et al. / Sensors and Actuators A 295 (2019) 59–69 Ziyu Zhao received her BS and MS degrees in Engineering Thermophysics from Northwestern Polytechnic University, China, in 2013 and 2016, respectively. She is working toward a Ph.D. degree in Engineering Thermophysics at Northwestern Polytechnic University. Her research interests include sensors for dynamic gap measurements. Yaguo Lyu is an associate professor of School of Power and Energy at Northwestern Polytechnic University, China. He received his BS, MS and PhD degrees in Power

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Engineering, all from Northwestern Polytechnic University. His research interests include lubrication engineering and sensors for dynamic gap measurements. Zhenxia Liu is a professor of School of Power and Energy at Northwestern Polytechnic University, China. He received his BS, MS and PhD degrees in Power Engineering, all from Northwestern Polytechnic University. His research areas include lubrication engineering and sensors for dynamic gap measurements.