Design and parameter optimization of contactless vertical inductive angle sensor

Vacuum 169 (2019) 108865

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Design and parameter optimization of contactless vertical inductive angle sensor

T

⁎

Zhipeng Lia,b, Chao Zhanga,b, , Songzhuo Shia, Xu Menga, Bonan Wanga a b

College of Transportation, Northeast Forestry University, Harbin, 150000, China Lishengda Mechanical and Electrical Technology Co., Ltd., Harbin, 150000, China

A R T I C LE I N FO

A B S T R A C T

Keywords: Contactless vertical inductive angle sensor Particle swarm optimization Measurement accuracy error

Contactless inductive angle sensors are widely used in the industrial field. To date, the contactless inductive angle sensor is a planar structure, and its physical structure limits the number of turns of the excitation coil, resulting in a small eddy current on the rotor, which increases the measurement accuracy error of the sensor. Therefore, in view of reducing the sensor volume and reducing the sensor measurement accuracy error, a contactless vertical inductive angle sensor is designed. Through modeling and simulation, when the position of the rotor and stator changes, the induced voltage in the receiving coil changes sinusoidally. The structural parameters that have the greatest influence on the measurement accuracy error of the sensor are obtained by simulation. Because of the complex electromagnetic structure of the sensor, the linearly-decrease inertia weight particle swarm optimization (LIWPSO) and finite element method (FEM) are used to optimize the sensor measurement accuracy error. In the simulation, the measurement accuracy error of the optimized sensor is 0.778% in the angle range of 0°–40°. The sensor is manufactured and tested for the optimized parameters, and the observed experimental measurement accuracy error is 0.981% in the angle range of 0°–40°.

1. Introduction Angle sensor is a sensor that converts the angle of an object to be measured into an electrical signal. It is widely used in various fields of modern industry. The angular displacement sensors can be classified into contact type and contactless type according to the measurement principle. Traditional contact sensors are mainly strained gauge and potentiometer type, the main disadvantage is that long-term usage and vibration can cause considerable wear [1], and as a result, noise in the signal is high and accuracy is reduced. Other sensor technologies include contactless sensors, which are mainly optical sensors [2], capacitive effect systems [3] and magnetic sensors [4,5]. Their advantages are fast response times and high precision, but their structures are generally complex, the processing is difficult, and the environmental requirements are strict. Electromagnetic inductive sensors have the advantages of a simple structure, low cost, strong anti-interference ability and no need for temperature compensation [6–8]; in these ways, it is an ideal sensor. The measurement accuracy error in these sensors varies with the physical structure of sensors [9]. Research on electromagnetic inductive sensors has focused on planar structures [10–12]. The receiving coil

and the exciting coil of the sensor of the planar structure are in the same plane, so that the number of exciting coil turns and the shape and number of the receiving coil are limited by the volume of the sensor. If the sensor is large in volume, the space air gap is large, which increases the measurement accuracy error. And compared with the constant magnetic field produced by a permanently magnetic material, the electromagnetic inductive sensor has a high-frequency alternating magnetic field, and the existence of higher-order harmonic signals ultimately leads to a large measurement accuracy error [13]. Contactless electromagnetic inductive sensors are widely used in various environments. In a vacuum, such as a space station or satellite, a measurement system is needed to measure the torque and angle of rotation of the shaft. The principles of electromagnetic induction and the principle of eddy current effects are equally applicable in a vacuum [14]. To ensure that the sensor can be operated in a vacuum, the material selection is a Flexible Printed Circuit (FPC), and the FPC manufacturing process includes vacuum curing [15]; in terms of electronic component selection, the selected electronic components have no special requirements for air pressure and height. This ensures that the sensor designed in this paper can be operated in a vacuum environment. The non-contact measuring system [16] for the micro-thrust measurement under vacuum

∗

Corresponding author. College of Transportation, Northeast Forestry University, Harbin, 150000, China. Tel.: +8615663712907, +8682191004 E-mail addresses: [email protected] (Z. Li), [email protected] (C. Zhang), [email protected] (S. Shi), [email protected] (X. Meng), [email protected] (B. Wang). https://doi.org/10.1016/j.vacuum.2019.108865 Received 20 June 2019; Received in revised form 4 August 2019; Accepted 9 August 2019 Available online 11 August 2019 0042-207X/ © 2019 Elsevier Ltd. All rights reserved.

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excitation coil and two groups of receiving coils output signals together to complete an angle measurement. According to the principle of electromagnetic induction, when the excitation coil is driven by a sinusoidal voltage [27,28], an alternating magnetic field is generated in the vicinity of the rotor, which causes eddy currents in the rotor [29]. Under the action of the alternating magnetic field and eddy current field, an induced voltage is generated in the receiving coil, as described by equation (1).

conditions in the thrust frame designed in Ref 16 can use the sensor designed in this paper. In order to solve the volume limitation problem and reduce the measurement accuracy error, a contactless vertical inductive angle sensor was designed. In order to reduce the measurement accuracy error, the sensor structure needs to be optimized. In recent years, researchers have begun using various adaptive algorithms and algorithms based on the swarm intelligence in order to obtain an optimal solution through collaboration and information sharing. For example, genetic algorithms and particle swarm optimization have been used to optimize certain sensor structural parameters [17,18], as well as responsive surface methodologies [19]. Particle swarm optimization (PSO) is a generalized swarm intelligence method for solving global optimization problems [20]. Compared with the traditional algorithm, the PSO is very fast and the global search ability is also very strong. After years of research, the particle swarm algorithm has made great progress. Among them, the multi-objective particle swarm optimization algorithm is the latest research direction [21]. In this paper, the PSO and FEM joint methods are used to optimize the sensor, so single-objective particle swarm optimization is used. Although the PSO has obvious advantages compared with other intelligent algorithms, it still suffers the problems of premature convergence and can potentially converge at a locally optimal solution. For this reason, Shi and Eberhart proposed a modified algorithm, which introduced inertia weight into the particle swarm optimization algorithm [22]. Inertia weight, ω, plays a decisive role in the convergence rate of a function. Determining the ratio of the local search ability to the global search ability is critical. Shi and Eberhart adopted the linearly decreasing strategy of inertia weight in order to coordinate the balance between global and local search abilities of the algorithm [23]. The inertia weight decreases with the number of iterations, which guarantees the global search ability of the algorithm and avoids falling into the local optimal solution. In this paper, the structural parameters of the sensor are optimized by combining the linearly-decrease inertia weight particle swarm optimization (LIWPSO) and finite element method (FEM) to obtain the structural parameters of the minimum measurement error. The structure of this paper is as follows: The second section introduces the structure and the working principle of the contactless vertical inductive angle sensor. The third section simulates the design model to verify the theoretical feasibility. The fourth section chooses the key design parameters and optimizes the sensor using LIWPSO-FEM. The results are measured and discussed in Section 5. The discussion is in the sixth section. Finally, the conclusion is drawn in the seventh section.

U=

d∫ (BE (t , x , y, z ) + BR (t , x , y, z )) dA dφ , = dt dt

(1)

In equation (1), BE is the alternating magnetic field produced by the exciting coil; BR is the eddy current field; and, A is the coupling area between the rotor and the receiving coil. The receiving coils arrangement is a positive and negative loop structure. While the induced voltage of two adjacent coils under the influence ofBE is the same, their direction is opposite. The net effect of BE on the induced voltage in the receiving coil is 0. Only the induced current of the eddy current field acting on the receiving coil is effective, as shown in equations (2) and (3).

d∫ (BE (t , x , y, z ) dt

= 0,

d∫ BR (t , x , y, z ) dA dφ = dt dt

(2)

(3)

Angle measurement is the process of angular displacement of the rotor relative to the receiving coil. In the course of displacement measurement, as can be seen from equation (4). When the magnetic field is constant, the magnetic flux is proportional to the coupling area of the rotor and the closed-loop of the receiving coil. The coupling process between the rotor and the receiving coil is shown in Fig. 2.

φ=

∫ B·dA,

(4)

During the rotation of the rotor from position 1 to position 2, the change of magnetic flux in the closed-loop of the receiving coil is as follows: φ → 0 → φ → 0 . Starting from the complete coupling of the rotor and the receiving coil, the mathematical model of the coupling region change is shown in equation (5) during the rotation of loop 1. 2 ⎧ A = S − 2(Δθ) tan α     Δθ ∈ (0,10) , 2 ⎨ ⎩ A = 2(20 − Δθ) tan α − S     Δθ ∈ (10,20)

(5)

Where S is the receiving coil closed-loop area; and 2α is the angle between the side lines of a single rhombic loops of the receiving coil; A is the coupling area; Δθ is the angular displacement. When the rotor and the receiving coil loop 1 are fully coupled, the coupling area between the rotor and the receiving coil loop 2 is 0, the position is calibrated as 0°, and the direction of the induced voltage of the receiving coil is defined as positive. At this time, the coupling area is the largest, the magnetic flux is the largest, and the induced voltage of the receiving coil is the largest. With the rotor rotates, the coupling area between the rotor and the receiving coil loop 1 decreases, and the coupling area between the rotor and loop 2 increases. When the rotor turns to 10°, the coupling area between the rotor and the receiving coil loop 1 and the receiving coil loop 2 is the same, the magnetic flux is the same; the induced voltage generated by the two loops is the same, but in the opposite direction, so the induced voltage of the receiving coil is 0. When the rotor rotates to 20°, the rotor and the receiving coil loop 2 are fully coupled, and the coupling area between the rotor and the receiving coil loop 1 is zero, the magnetic flux is the maximum, and the induced voltage of the receiving coil reaches the maximum in the negative direction. When the rotor rotates from 20° to 40°, the induced voltage of the receiving coil decreases from a negative maximum to zero and then increases to a positive maximum. This process is a measurement period. Assume that the length and width of a single

2. Theoretical research on a contactless vertical inductive angle sensor In order to increase measurement accuracy, the contactless vertical inductive angle sensor designed as a symmetrical master-slave structure consisting of a stator and two rotors. The stator is designed with a FPC wiring. The FPC features lightweight, thin thickness, and bendability, due to the fact of polyimide be used as a thin plate and being very flexible [24]. Their reduced thickness and ability to bend and adapt to various shapes are important to the sensor design [25]. The stator consists of an excitation coil and two groups of receiving coils. Each receiving coil group has three receiving coils, and the difference between adjacent receiving coils is 6.67°. Each receiving coil is arranged as a single conductor connected by 18 diamond rings, with a single diamond ring occupying 20°. It has been observed that rectangular shapes contribute to better sensor performance and linearity [26], so the rotors are nine rectangular aluminum sheets 20° apart. The excitation coil is a spiral coil. The main stage of the sensor is used for measurement and the slave stage is used for calibration. This structure is shown in Fig. 1, where the main stage is composed of rotor A and receiving coil group 1, and the secondary stage is composed of rotor B and receiving coil group 2. The two groups of receiving coils share one 2

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Fig. 1. Integral structure diagram of the contactless vertical inductive angle sensor.

rhombic loop is 0.75 mm and 1 mm respectively. A curve describing the relationship between coverage area and angular displacement is obtained by MATLAB simulation and shown in Fig. 3. The fitting equation is S = 0.375 sin(π /20 × θ) , that is to say, the change of the coupling area is equivalent to a sinusoidal curve. Therefore, the sinusoidal variation of the coupling area between the rotor and the receiving coil indicates that the output voltage of the sensor must also be sinusoidal. The period of a sinusoidal signal is 40°, so the period of angle measurement, therefore, is 40°. When measuring with a receiving coil, the output voltage of the receiving coil is easily disturbed by sensor manufacturing and assembly errors, which result in low measurement accuracy. Moreover, a receiving coil cannot linearize the angle signal. So, three receiving coils are used in this design. In the structure, the three receiving coils are designed as a staggered arrangement offset 6.67° from each rhombic coil. The sinusoidal output voltage of the three receiving coils is illustrated in Fig. 4.

Fig. 3. The relationship between the coupling area of a rotor and receiving coil and the angular displacement.

ANSYS MAXWELL 3D in order to solve for the eddy current and perform transient analysis. The meshing of the 3D model is illustrated in Fig. 5. The maximum length of the internal elements is 2 mm, and the total number of meshes is 79615. The mesh partitioned by the 3D model is tetrahedral. There are three ways to load excitation current by ANSYS Maxwell, input by constant or function, reference to tabular data through the Design Datasets command and loading external circuit. The method of loading the external circuit is adopted in this paper. The Maxwell circuit editor is used to plot the circuit diagrams of the excitation coils and three groups of receiving coils under transient field dynamic analysis, as shown in Fig. 6. The excitation coil power supply uses 5 V alternating current with a frequency of 4 MHz and an internal resistance of 1Ω. The

3. Simulation analysis of contactless vertical inductive angle sensor 3.1. Static analysis of transient field The sensor designed in this paper is based on electromagnetic induction between the rotor and stator in order to achieve an angle measurement. It is first necessary to analyze the alternating magnetic field produced by the excitation coil under sinusoidal excitation. It is obtained from the previous analysis that the induced voltage in the receiving coil changes periodically as the rotor rotates. Using SOLIDWORKS, a three-dimensional (3D) model is built and imported into

Fig. 2. Coupling process between the rotor and receiving coil. 3

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Fig. 4. Sinusoidal variation of induced voltage in three receiving coils.

Fig. 7. Spatial distribution of alternating magnetic field and eddy current field.

Using the ANSYS MAXWELL 3D eddy current solver, the eddy current on the rotor is shown in Fig. 7. From the pictures, when the simulation time is 1.3e-007s, the eddy current of the rotor is about 3.48e+005 A_per_m2. The relative position of the rotor and stator remains unchanged, the induced voltage in the receiving coils is shown in Fig. 9. The induced voltage is a sinusoidal signal. The magnetic field was unstable at first, so the simulation started at 0.7us. At a fixed position, the amplitude of the induced voltage of each receiving coil is constant. As can be seen from Fig. 8, the induced voltage of the three receiving coils has different amplitudes, but the frequency is the same. The induced voltage period is about 250 ns and the corresponding frequency is 4 MHz. The frequency is the same as that of the excitation coil.

3.2. Dynamic analysis of transient field After the static analysis, dynamic analysis is performed. The induced voltage of the three receiving coils is analyzed when the rotor rotates. Using ANSYS MAXWELL 3D it is possible to solve for the transient motion state, while having the rotor rotate around the Z-axis in a band domain configuration. The change in induced voltage in the receiving coil can be observed with the rotation of the rotor. The total simulation time is 17.5μs with a step size of 0.02 μs. The induced voltage in the receiving coil is simulated and shown in Fig. 9. When the rotor rotates at a constant angular speed, the amplitude of the induced voltage in the receiving coil depends on the relative position of the rotor. Therefore, the voltage signal induced consists of the carrier signal plus the modulation signal. The carrier signal is sinusoidal with the same frequency as the excitation. The modulated signal is a bilateral sinusoidal signal with a period of 40°. The co-simulation of three receiving coils is shown in Fig. 10. Again, the induced voltage is a sinusoidal signal. The three sets of signals are independent of each other, with the same amplitude, frequency but phase offset of 40°. The simulation results are consistent with theory, which is considered as verification of the accuracy of the theory. It can be seen from the simulation diagram that the output signal of the system is three high frequency modulated signals with a certain phase difference. Therefore, it is necessary to perform detection, amplification, and digital-to-analog conversion processing on the highfrequency oscillation signal to obtain a sinusoidal signal that satisfies the measurement requirement. This content is described in a subsequent article.

Fig. 5. 3D modeling of studied sensor under mesh operation.

Fig. 6. Coil external voltage in transient field solver.

internal resistance of the three groups of receiving coils is set at 0.5 Ω each. According to the design theory, under a high-frequency sinusoidal excitation, an alternating magnetic field is generated around the excitation coil. The total simulation time is 4μs with a step size of 0.005μs. 4

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Fig. 8. Induced voltage in the receiving coils (static analysis).

Fig. 9. Induced voltage in 3 receiving coils (dynamic analysis).

Fig. 10. The co-simulation of three receiving coils.

5

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Fig. 11. Selection of voltage signal section.

3.3. Signal processing algorithms

Table 1 40° offset output voltage algorithm formula. Measurement section

Calculation formula

1 2 3 4 5 6

UO UO UO UO UO UO

= = = = = =

The ideal input and output characteristics of the sensor are linear. It has many advantages: simplifying the analysis and design calculation of sensors; easy calibration; avoiding non-linear compensation. In the practical application of sensors, the instantaneous displacement of the rotor varies randomly. Therefore, the linear relationship between the output voltage and the angular displacement of the rotor needs to be obtained by algorithm. According to the characteristics of the sinusoid, the area near zero is approximately linear. The three signals received in the receiving coil are named A, B, and C, respectively. After storing these three sets of signals, then reverse them, and three additional sets of signals are obtained, and they are named A , B and C ; in total there are six sinusoidal signals. The algorithm used to properly sort the received signals is depicted in Fig. 11. The line segment between UMAX and UMIN is selected as the measurement segment for analysis. After determining the selected measurement signal, according to the signal segment in which it is located, the output voltage is obtained by processing according to the formula in Table 1. The inducted voltage is linearly related to the angular displacement of the rotor during the measurement period, as shown in Fig. 12.

UA UC + (UMAX − UMIN ) UB + 3UMAX − UMIN U A + 5UMAX − UMIN UC + 7UMAX − UMIN U B + 9UMAX − UMIN

UO —Linear output voltage.

4. LINPSO-FEM for structural parameters of contactless vertical inductive angle sensor After obtaining the linear relationship between the angle and the voltage of the electromagnetic coupling system, it is necessary to analyze its measurement accuracy error. The measurement accuracy error is expressed by dividing the maximum error between the ideal angle and the measured angle by the measurement period. Expressed by formula (6):

Fig. 12. Linear relation between angle and voltage of electromagnetic coupled system. Table 2 Input parameters in sensor model.

|ϕi − ϕ|max × 100%, 40

Parameter

Initial Value

Valid Range

L=

Excitation coil turns number Rotor thickness Rotor radius Receiving coil width

10 0.5 mm 14.5 mm 0.2 mm

4–12 0.2 mm–0.6 mm 14.1 mm–14.5 mm 0.2 mm–0.35 mm

Where L is the measurement accuracy error of the sensor, ϕ is idealized angle, ϕi is simulation or measured angle. From the above analysis, the measurement accuracy error of the sensor is affected by the rotor and the stator, which include the number of excitation coil turns, rotor thickness, rotor radius and receiving coil width. In order to simplify the design calculation, the key variables are 6

(6)

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Table 3 Sensor measurement accuracy error. φ (o)

U1 (mV)

U2 (mV)

U3(mV)

0 5 10 15 20 25 30 35 40

0.0110 2.1920 3.2500 2.2417 0.1127 −2.3595 −3.2460 −2.3605 0.0212

2.8091 0.9821 −1.6398 −3.1587 −2.8692 −0.7759 1.7632 3.1149 2.8039

−2.8201 −3.1740 −1.6101 0.9170 2.7565 3.1153 1.4928 −0.7545 −2.8251

U1-Induced

voltage of receiving coil1,

U 2 -Induced

voltage of receiving coil2,

U3 -Induced

UO (mV) 0.0110

2.2657 4.8567 7.3383 9.5796 12.1933 14.6471 16.8712 19.3994

voltage of receiving coil 3,

ϕi -Simulation

φi (o)

φi − φ (o)

L(%)

0.0216 4.7124 10.0336 15.1544 19.7792 25.1724 30.3172 34.8248 40.0416

0.0216 −0.2876 0.0336 0.1544 −0.2208 0.1724 0.3172 −0.1752 0.0416

0.054 −0.719 0.084 0.386 −0.552 0.431 0.793 −0.437 0.104

angle,L -Measurement accuracy error.

Table 4 Effect of the number of excitation coil turns on sensor measurement accuracy error. Excitation coil turns number

L

4 6 8 10 12

0.858% 0.846% 0.829% 0.793% 0.831%

Table 5 Effect of rotor thickness on sensor measurement accuracy error. Rotor thickness

L

0.2 mm 0.3 mm 0.4 mm 0.5 mm 0.6 mm

0.967% 0.879% 0.801% 0.793% 0.799%

Fig. 13. Fitness value with iteration numbers.

Table 6 Effect of rotor radius on sensor measurement accuracy error. Rotor radius

L

14.1 mm 14.2 mm 14.3 mm 14.4 mm 14.5 mm

0.827% 0.819% 0.799% 0.795% 0.793%

Table 7 Effect of receiving coil thickness on sensor measurement accuracy error. Receiving coil width

L

0.20 mm 0.25 mm 0.30 mm

0.793% 0.802% 0.814%

Fig. 14. Measurement accuracy error vs. the rotor thickness obtained in optimization design. Table 9 Optimized sensor manufacturing parameters.

Table 8 Effect of receiving coil thickness on sensor measurement accuracy error. Parameter

Standard deviation

Excitation coil turns number Rotor thickness Rotor radius Receiving coil width

0.0245 0.0745 0.0154 0.0105

Optimal Variables

Measurement accuracy Error (%)

Rotor thickness 0.52 mm

L 0.778%

selected in the sensor design. In the simulation, in order to avoid extensive computation times, the excitation coil is generally simplified as a cylinder. In the manufacturing process, the total width of the excitation coil is set to 15 mils.

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Table 10 Comparison of rotor eddy currents between a traditional contactless inductive angle sensor and the contactless vertical inductive angle sensor. 1. Compared to the sensor designed in Ref 19, the structure is changed from a planar structure type to a vertical master-slave type structure, as shown in Fig. 1. There are three receiving coils in each stage, and a total of six receiving coils are associated with the measurement, which improves the measurement accuracy. Under the same excitation [19,30,33], the vertical structure has a larger rotor eddy current, as shown in Table 10. The structure of the master-slave stage can measure torque, which is the next step that needs to be perfected. 2. In terms of materials, the sensor designed in Ref 19 uses a conventional PCB. The sensor designed in this paper uses FPC. FPC has the advantages of high wiring density, lightweight and thin profiles [34]. It is suitable for vertical structure layout and reduces the overall size of the sensor. 3. Since the sensor size designed in Ref 19 is unknown, the best-selling HELLA sensor was compared to the sensor designed in this paper. Both the HELLA sensor and the sensor designed in Ref 19 are planar structures. As shown in Fig. 20, (a) is the main view, (b) is the top view. The Contactless vertical inductive angle sensor designed in this paper has a size of 35 × 35 × 40mm3 , the sensor radial maximum size is reduced by 50%, and the total volume is reduced by 39.6%. 4. In terms of maximum measurement accuracy error, the maximum measurement accuracy error of the sensor designed in this paper is 0.981%, which is higher than the maximum measurement accuracy error of 0.081% of the sensor designed in Ref 19. In addition to some of the differences in the algorithm, the main reason is the assembly error when the stator is fixed in the cover. The circle enclosed by the stator is not a standard circle, which leads to a large measurement accuracy error, which is the next step to be optimized. 5. Compared with the receiving coil of the polygonal structure designed in Ref 19, the receiving coil of the diamond ring structure designed in this paper has a simple structure, less parameter optimization, more coils, and higher theoretical precision. 6. In terms of angle algorithm, the angle algorithm designed by Ref 19 can't calculate the angle when 0 mV is collected in the receiving coil. The angle algorithm used in this paper does not have any defect. According to the principle of the vernier caliper [35], the measurement range can be increased to −360°~360°, which is currently under study. 7. In terms of optimization algorithm, LINWPSO has faster computational speed than traditional PSO used in Ref 19, saves computing resources, and is not easy to fall into a local optimal solution. Parameter

Traditional Contactless inductive Angle Sensor

Contactless Vertical Inductive Angle Sensor

Excitation frequency Eddy current on the rotor(A_per_m2)

10 * sin (2 * pi * 1000000 * time) 1.031e+007

10 * sin (2 * pi * 1000000 * time) 1.5005e+008

4.1.1. Excitation coil turns number The induced voltage in the receiving coil is next simulated as a function of the number of excitation coil turns. Rotor thickness, rotor radius and receiving coil width are initial values, Table 4 describes a progressive increase in the number of turns of the excitation coil from 4 to 12. It is shown that the number of turns of the excitation coil has little influence on the measurement error of the sensor, as the maximum and minimum measurement accuracy error has a variation of 0.065%. 4.1.2. Rotor thickness The thickness of the rotor affects the eddy current in the rotor, and therefore the induced voltage in the receiving coil. Excitation coil turns number, rotor radius and receiving coil width are initial values, Table 5 describes the thickness of the rotor varies from 0.2 mm to 0.6 mm for a fixed number of excitation coil turns. The variation of simulated measurement accuracy error has a variation of 0.174%, which has a great influence on the accuracy of the sensor.

Fig. 15. Measurement accuracy error of the simulation model.

4.1. Sensor parameter selection 4.1.3. Rotor radius Generally, the larger the radius of the rotor, the smaller the distance between the stator and rotor, and the larger the eddy current of the rotor. Excitation coil turns number, rotor thickness, and receiving coil width are initial values, Table 6 describes the influence of rotor radius on the variation of measurement accuracy error of the sensor, calculated to be 0.034%.

The induced voltage is theoretically an ideal sinusoidal curve, therefore a linear relationship between the phase angle and the angular displacement over the period should be obtained. However, in practice, the observed relationship between phase angle and angular displacement is nonlinear due to manufacturing and assembly errors, as well as nonlinear eddy current effects [30]. The main design parameters of the sensor include the excitation coil turns number, rotor thickness, rotor radius and receiving coil width. The sensor model is drawn according to the initial value, and the method of control parameters are used to select the structural parameters that have the greatest influence on the measurement accuracy error. The initial design parameters and ranges are shown in Table 2. Using the initial input parameters of the sensor model, a finite element simulation is performed. The induced voltage in each group of receiving coils is obtained and the measurement accuracy error is calculated. Table 3 shows that the maximum error between simulated and theoretical rotation values in the 30° is 0.3172°. The maximum measurement accuracy error of the sensor is 0.793% in the range of 0°–40°.

4.1.4. Receiving coil width Based on the inductance calculation manual [31], the width of the coil has little effect on coil inductance and impedance. Due to the limitations of current processing technology, the minimum line width that can be reliably manufactured is 6 mils, i.e., 0.153 mm. Generally speaking, the line width is set to 8 mils, i.e., 0.203 mm. The simulation model uses 0.2 mm as the initial coil width parameter to observe the influence of the width of three groups of receiving coils on the average sensor measurement accuracy error. As shown in Table 7, the measurement accuracy error has a variation of 0.021% shows that the coil width has little effect on the measurement accuracy of the sensor. 4.1.5. Optimization parameter selection The standard deviation represents the magnitude of the fluctuations 8

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Fig. 16. Manufactured FPC and sensor for mounting the housing.

sensor measurement accuracy error. 4.2. Sensor optimization In this paper, the structural parameters of the sensor are optimized by combining the linearly-decrease inertia weight particle swarm optimization (LIWPSO) and the finite element method (FEM). LIWPSO depends on two factors, i.e., the change of particle position and velocity [32]. In the search space, each particle has a position that represents a possible solution, that is, a position represents a set of sensor structural design parameters. LIWPSO finds the best design variable by solving the optimal value of fitness function in the region. Based on this variable, the sensor model is designed, and the sensor is simulated by the finite element method analysis. The induced voltage is then also obtained by simulation, and the corresponding angle is calculated. The angle value is compared with the ideal line, and the sensor measurement accuracy error is calculated. That is to say, finding a group of sensor design parameters can make the sensor generate minimum measurement accuracy error. The adaptive process of the LIWPSO is shown in formulae (7), (8), and (9):

vij (t + 1) = ωvij (t ) + c1 r1 (t )(pij (t ) − x ij (t )) + c2 r2 (t )(pgi (t ) − x ij (t )),

Fig. 17. Experimental sensor. 1-Display device of Goniometer equipment, 2-Monitor, 3-Motor, 4-Contactless vertical inductive angle sensor, 5-DAQ board, 6-Oscilloscope.

(7)

x ij (t + 1) = x ij (t ) + vij (t + 1),

in the data set. Calculate the standard deviation of the measurement error of the excitation coil turns number, rotor thickness, rotor radius and receiving coil width. Find the parameters corresponding to the largest standard deviation and optimize. The standard deviation of each parameter is shown in Table 8. In summary, when the rotor thickness changes from 0.2 mm to 0.6 mm, the standard deviation of the measurement accuracy error is 0.0745. Compared with simulated variations in other parameters, the rotor thickness has the greatest influence on the measurement accuracy error of the sensor. The rotor thickness is the key parameter affecting

ω = ωmax

t·(ωmax − ωmin ) , tmax

(8)

(9)

In these equations, ω is the inertia weight, c1 and c2 are the acceleration coefficients, ωmax and ωmin are the maximum and minimum allowed values of inertia weight, and tmax is the maximum number of iteration steps. Larger values of ω correspond to strong global convergence and weak local convergence, whereas small values of ω correspond to strong local convergence and weak global convergence. Generally,ωmax = 0.9 and ωmin = 0.4. 9

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Fig. 18. Induced voltage, A and B, in two receiving coils.

Fig. 19. Measurement accuracy error of the sensor.

Fig. 20. Contactless vertical inductive angle sensor compared with HELLA sensor.

In the LIWPSO-FEM, the fitness function is used to represent the measurement accuracy error. The rotor thickness is taken as the design parameter. By searching for the minimum of the fitness function, the optimal design of the sensor is transformed into the problem of determining the value of sensor variables. The expression of the fitness

function is described in equation (10):

fitness = L =

|θi − θ| × 100%, 40

(10)

Where the rotor thickness is constrained to 0.2 mm–0.6 mm. L is the sensor measurement accuracy error,θ is the rotation angle of the rotor, 10

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θi is an idealized rotor rotation angle fitted from a straight line. When the LIWPSO-FEM is used to optimize the sensor, with the number of particles set to 20, acceleration coefficients c1 = c2 = 1.4945, and the speed limit, vij, constrained to [-0.05–0.05]. The number of iterations is set to 30. The minimum fitness value is obtained by iterative calculation, and the optimal structural parameters of the rotor are obtained. After the iteration process is completed, the fitness function iteration process is shown in Fig. 13. When the number of iterations exceeds 18, the minimum linearity is stable at 0.778%. The corresponding thickness of the rotor is 0.52 mm. The measurement accuracy error and the rotor thickness obtained in the optimization process are plotted in Fig. 14. By optimizing the thickness of the rotor, the optimal manufacturing parameters of the sensor are obtained. The optimized parameters are shown in Table 9. When the thick of the rotor approximate 0.52 mm, the measurement accuracy error is 0.778% in Fig. 15.

where high sensor performance is required with respect to resolution, measurement speed, stability and reliability, and this performance has to be achieved with very limited power consumption (in satellites) and with limited means for removing the generated heat (in vacuum environment). The next research focuses on how to reduce the power consumption and reduce the eddy current thermal effect of the rotor when the contactless vertical inductive angle sensor is applied in a vacuum environment. 7. Conclusions A contactless vertical inductive angle sensor is designed in this paper. According to the principle of electromagnetic induction, a contactless vertical inductive angle sensor composed of a stator and a rotor is designed. The stator consists of an excitation coil and two sets of receiving coils (each set of three receiving coils). The simulation shows that the coupling process of the rotor and the receiving coil is sinusoidal. According to this phenomenon, the difference between adjacent receiving coils is designed to be 6.67°. The excitation coil is excited by a sinusoidal excitation at a frequency of 4 MHz, the induced voltage in the receiving coil varies sinusoidally with the angular position of the rotor and stator. Through ANSYS MAXWELL software, the sensor is modeled and simulated, the influences of rotor thickness, rotor radius, excitation coil turn number and receiving coil width on the sensor measurement accuracy error is analyzed. LIWPSO and FEM to calculate the fitness of each rotor thickness, and the minimum fitness is found when the rotor thickness is 0.52 mm; the simulated measurement accuracy error in the rotor rotation range from 0° to 40° is 0.778%. A sensor is then manufactured with this parameter and the experimental angle measurements of 0°–40° show that the observed measurement accuracy error is 0.981%. In the inductive angle sensor field, the allowable measurement accuracy error is about 1% [37,38], therefore this design is well within industry standards.

5. Fabrication and testing of the sensor To complete the research, the board was designed according to the optimized parameters, and the circuit board was fabricated using a flexible material, that is, FPC. Compared with the conventional Printed Circuit Board (PCB), the FPC has the smaller pitch, line width, and line spacing, so the manufacturing precision is higher. In order to make the sensor have a smaller volume, the signal processing circuit portion and the stator are designed on the same FPC, as shown in Fig. 16. The front of the FPC is shown in Fig. 16(a), and the reverse side of the FPC is shown in Fig. 16(b).Mount the FPC on the sensor housing as shown in Fig. 16(c), (d). In order to verify the correctness of the above theory, the following experiments are carried out, as illustrated in Fig. 17. The angular displacement experiment is mainly tested by the Goniometer equipment. According to the angle instruction sent by an industrial computer, the motor rotates the corresponding angle according to the received instruction. The stator of the sensor to be measured is coaxially fixed with the motor, and the rotor rotates along the axis. The processing circuit of the sensor deals with the measured signal and sends the processed signal to the monitor through the DAQ board to get the measured angle. The oscilloscope is connected with the sensor to observe the corresponding signal changes. The probe of the oscilloscope is placed at the Pulse Width Modulation (PWM) output port of the sensor. When the rotor rotates, the oscilloscope is used to test the inductive voltage of the A and B receiving coils (the oscilloscope can only test two signals at a time) as shown in Fig. 18. It can be seen from the figure that the voltage of the receiving coil is a sinusoidal signal. The two groups of voltage signals have the same amplitude and frequency and have different phase angles. The Goniometer equipment is used to rotate the measuring axis from 0° to 40° and read the results of the sensor measurement. Comparing the measured value with the theoretical value, the nonlinear error is calculated according to formula (6). The result is shown in Fig. 19. According to the experimental data, the measurement accuracy error of the sensor is 0.981%. The observed measurement accuracy error is larger than the simulated data, which is assumed to be due to manufacturing errors, and subsequent re-optimization is recommended.

Author contributions Chao Zhang designed and implemented the sensor, discussions and preparing the manuscript. Zhipeng Li contributed to performing of the experiments and discussions. Songzhuo Shi designed signal process circuit. Bonan Wang designed the physical angular displacement simulation experiments. Xu Meng gave me some advice about the paper structure arrangement. Acknowledgments This research was financially supported by grants from the National Science Foundation of China (Grant Number: 51575097), the Fundamental Research Funds for the Central Universities (No.2572019CP04). References [1] Akira Noguchi, Kosuke Yamawaki, Toshiro Yamamoto, Development of a steering angle and torque sensor of contact-type, Furukawa Rev. 25 (2004) 36–41. [2] R.D. Evans, N.M. Jokerst, R.B. Fair, Integrated optical sensor in a digital microfluidic platform, IEEE Sens. J. 5 (2008) 628–635. [3] M. Gasulla, X. Li, C.M. Gerard, Meijer, A contactless capacitive angular-position sensor, IEEE Sens. J. 3 (2003) 607–614. [4] E. Hristoforou, P.D. Dimitropoulos, J. Petrou, A new position sensor based on the MDL technique, Sens. Actuators A Phys. 132 (2006) 112–121. [5] B. Legrand, Y. Dordet, J.-Y. Voyant, J.-P. Yonnet, Contactless position sensor using magnetic saturation, Sens. Actuators A Phys. 106 (2003) 149–154. [6] O. Sosnicki, G. Michaud, F. Claeyssen, Eddy current sensors on printed circuit board for compact mechatronic application, Proceedings of Sensoren und Messsysteme, Meylan, France, 18–19 May, 2010, pp. 244–251. [7] Z. Zhang, J. Zheng, H. Wu, Development of a respiratory inductive plethysmography module supporting multiple sensors for wearable systems, Sensors 12 (2012) 13167–13184. [8] L. Huang, A. Rahman, W.D. Rolph, et al., Electromagnetic finite element analysis for designing high frequency inductive position sensors, IEEE Trans. Magn. 37 (5)

6. Discussion This paper compares with the research results of the predecessors (mainly compared with the results of Ref 19), and the achievements and shortcomings are as follows: The application of the sensor is in a special vacuum environment with severe restrictions of power dissipation and increased requirements for reliability and thermal drift [36]. There are applications 11

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