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Systematic Theoretical Analysis of Dual-Parameters RF Readout by a Novel LC-Type Passive Sensor.

1. Introduction

Owing to their characteristics of wireless power supply and signal readout, various wireless passive sensors have been developed for harsh environments such as high-temperature [1-3] and in vivo environments [4, 5] in the past decade. However, most currently reported passive wireless sensors are single-parameter sensors, making it difficult for them to meet simultaneous multiparameter measurement requirements, such as pressure and temperature monitoring in turbine engines [6]. In addition, measuring multiple parameters by multiple single-parameter sensors will certainly deteriorate the installation adaptability and bring more intrusive interference. Therefore, developing micro passive sensors that can simultaneously measure multiple parameters has great significance for environmental monitoring in harsh environments.

Catering to the above-mentioned multiparameter measurement requirement, Zhang et al. proposed a concept of integrating two LC resonators into a single sensor to realize dual-parameters RF readout firstly (shown in Figure 1) [7]. Actually, the principle of this design was also used to enhance the signal strength of single-parameter passive sensors previously [8, 9]. Owing to the large overlapped area of the inductance coils of the sensor, its size does not increase significantly compared to the single-parameter counter-part. However, large overlapped area of inductances coils results in strong mutual coupling which make the crosstalk among multiple parameters nonignorable. Up to now, there is no scheme proposed to decouple the crosstalk. In order to avoid the crosstalk, Zhang et al. proposed a specific-winding inductance coils to suppress the mutual coupling between the sensor coils in their later paper [10], thus making the crosstalk negligible. Although the specific-winding method was quite successful in suppressing the crosstalk caused by the mutual inductance, this method makes the inductance of sensor coils decreased heavily, which in return shortens the readout distance. The fact is that there is no paper systematically studied influencing factors of dual-parameters RF readout by the LC-type passive sensor, as well as the crosstalk decoupling scheme.

In order to provide guidelines for optimum design of the dual-parameter LC-type passive sensor system, problems of signal strength of the sensor and crosstalk between dip frequencies were chosen to be theoretically studied and discussed in this paper. More importantly, this paper proposed a decoupling function to solve the crosstalk problem successfully.

2. Analysis Model

Lumped circuit model of the LC type passive dual-parameter sensor system is illustrated in Figure 2, and the sensor is equivalent to two LC resonant circuits. [L.sub.1] and [L.sub.2] are the inductors of the sensor, [R.sub.1] and [R.sub.2] are the series resistances of the sensor, and [C.sub.1] and [C.sub.2] are the sensitive capacitors of the sensor. Similarly, the readout coil is equivalent to an inductor [L.sub.a] and a series resistance [R.sub.a]. In order to realize multiparameter measurement by a micro sensor with smallest volume, the inductance coils of the sensor usually have a large overlap area, which results in the existence of mutual inductance [M.sub.12]. When the sensor was magnetically coupled with the readout coil, there are also mutual inductance [M.sub.1] and [M.sub.2], which corresponds to the coupling between the readout coil and the sensor inductors.

The input impedance of the readout coil can be given by [7]

[mathematical expression not reproducible] (1)

where [omega] is the angular frequency and is equal to 2[pi]f. And [k.sub.1] and [k.sub.2] are the coupling coefficients between the readout coil and the sensor coils. [k.sub.12] are the coupling coefficients between the sensor coils. [Z.sub.1] and [Z.sub.2] are the equivalent impedance of these two LC tanks, respectively, and they can be given as

[Z.sub.1] = [R.sub.1] + jw[L.sub.1] + 1/jw[C.sub.1], (2a)

[Z.sub.2] = [R.sub.2] + jw[L.sub.2] + 1/jw[C.sub.2].(2b)

The resonance frequency of these LC tanks can be written as

[f.sub.1] = 1/2[pi] [square root of ([L.sub.1][C.sub.1])], (3a)

[f.sub.2] = 1/2[pi] [square root of ([L.sub.2][C.sub.2])], (3b)

Similar to the readout of the single-parameter LC-type passive sensor, it is theoretically feasible that the resonance frequencies change of the dual-parameter sensor can also be detected by measuring the impedance parameters (e.g., phase, real part, and magnitude) of the readout coil. In order to obtain unambiguous characteristic frequencies which can represent the resonant frequencies of two LC tanks of the sensor, it is necessary to make the resonant frequencies [f.sub.1] and [f.sub.2] separated within the measurement range of the sensor when we design the sensor. In this paper, the inductances [L.sub.1] and [L.sub.2] are designed to be constant and the resonant frequencies changes of the sensor depended on the change of capacitances [C.sub.1] and [C.sub.2].

Herein, by substituting the parameters in Table 1 into (1) and using MATLAB software for plotting curve, the frequency-phase curve illustrated in Figure 3 was obtained. It is clear that there are two obvious phase dips when the sensor magnetically coupled with the readout coil. However, the phase dip frequencies ([f.sub.1min] = 39.56 MHz, [f.sub.2min] = 61.52 MHz in this design) are not equal to the resonant frequencies ([f.sub.1] = 41.09 MHz, [f.sub.2] = 56.27 MHz) of the sensor due to the mutual coupling [k.sub.12]. Specifically, the mutual coupling between the sensor coils makes the phase dip frequencies apart compared to the resonant frequencies; that is, the dip frequency [f.sub.1min] is smaller than the corresponding resonant frequency [f.sub.1] and [f.sub.2min] is larger than [f.sub.2].

In spite of the inequality between dip frequencies and resonant frequencies, dual-parameter measurement by tracking the dip frequencies is absolutely feasible because the change of the resonant frequency will make the corresponding dip frequencies change. As shown in Figure 4, the dip frequency [f.sub.1min] deceased monotonously when the capacitance [C.sub.1] increased from 15 pF to 19 pF (other parameters keep constant). It should be noted that the change of [f.sub.1] results in the drift of the dip frequency [f.sub.2min], and this phenomenon is a kind of crosstalk which is originally caused by the mutual coupling ([k.sub.12]) between the sensor coils. The crosstalk strength can be indicated by the drift value [DELTA][f.sub.2min] cross. Usually, the magnitude of the phase dip ([DELTA][[theta].sub.1] and [DELTA][[theta].sub.2] in Figure 2) should be large enough to make the readout of two phase dip frequencies possible within the measurement range.

3. Numerical Simulation and Analysis

3.1. Influencing Factors of the Sensor's Signal Strength. As illustrated in Figure 3, simultaneous dual-parameter measurement can be realized by tracking the dip frequencies [f.sub.1min] and [f.sub.2min]. Normally, passive sensors are developed for harsh-environment application, such as high-temperature and hermetic environment. And a typical situation is that the phase difference [DELTA][theta] corresponding to the resonant point of a passive sensor will decrease at elevated temperatures due to the degraded Q factor of the LC resonator, which will make tracking the frequencies [f.sub.1min] and [f.sub.2min] harder, and will eventually make it impossible for the sensor to work in higher-temperature environment. Therefore, it is quite important to optimize the sensor system design to achieve large possible phase difference [DELTA][[theta].sub.1] and [DELTA][[theta].sub.2] if other conditions such as readout distance and sensor dimensions permit. In order to provide guidelines for optimizing the sensor system to achieve considerable readout signal strength, the influencing factors of [DELTA][[theta].sub.1] and [DELTA][[theta].sub.2] need to be analyzed.

3.1.1. Quality Factor. As for a LC resonator, its Q factor can be given as

Q = [[omega].sub.0][L.sub.s]/[R.sub.5] = 1/2[pi][R.sub.s] [square root of ([L.sub.s]/[C.sub.s])]. (4)

It is obvious from (4) that the Q factor of the LC resonator mainly depends on its equivalent series resistance. Therefore, the resistance [R.sub.1] and resistance [R.sub.2] were changed to adjust the Q factors of the sensor in this part, and other parameters keep constant. As illustrated in Figure 5, the phase difference [DELTA][[theta].sub.1] increased obviously with the increase of Q factor [Q.sub.1], and [DELTA][[theta].sub.2] also showed a slight increase when [Q.sub.1] increased. It can be seen from Figure 6 that the increase of [Q.sub.2] resulted in an obvious increase of [DELTA][[theta].sub.2] and a slight increase of [DELTA][[theta].sub.1]. Overall, it is sure that the increase of Q factor will make the corresponding phase difference increase significantly and make the other one increase slightly. So the Q factors of the LC resonators should be increased as much as possible by the optimum design to enhance the signal strength of sensor if other conditions permit.

3.1.2. Coupling Coefficients. In the dual-parameter passive sensor system, there are three coupling coefficients [k.sub.1], [k.sub.2], and [k.sub.12]. As illustrated in Figure 7(a), the increase of [k.sub.12] not only made the phase difference [DELTA][[theta].sub.1] and [DELTA][[theta].sub.2] and changed, but also made the obvious drift of [f.sub.1min] and [f.sub.2min]. Specifically, [f.sub.1min] decreased and [f.sub.2min] increased when [k.sub.12] increased from 0.2 to 0.4 (other parameters keep constant). It can be seen from Figure 7(b) that the phase difference [DELTA][[theta].sub.1] increased from 39.9[degrees] to 43.8[degrees] when [k.sub.12] increased from 0.2 to 0.4, but [DELTA][[theta].sub.2] decreased from 14.6[degrees] to 4.6[degrees]. Overall, increasing the coupling coefficient [k.sub.12] makes phase difference corresponding to the small frequency point [f.sub.1min] increased slightly and makes the phase difference corresponding to [f.sub.2min] decreased obviously. In the dual-parameter passive sensor system, the signal strength corresponding to the larger frequency [f.sub.2min] was usually suppressed due to mutual coupling between the sensor coils. Therefore, considerable sensor strength can also be achieved by reducing [k.sub.12], which usually means to reduce the overlap area of the sensor coils in the sensor layout. However, many situations call for multiparameter sensor with volume as small as possible, which in return need the sensor coils overlapped as much as possible. So it is necessary to try to suppress the value of [k.sub.12] if other restrictions permit.

It is obvious from Figure 8 that change of the coupling coefficient [k.sub.1] has significant influence on the value of A01 and AQ2. When [k.sub.1] increased from 0.08 to 0.12, A01 increased from 31.3[degrees] to 54.6[degrees] and [DELTA][[theta].sub.2] decreased from 11.9[degrees] to 5.1[degrees] accordingly. Therefore, the increase of the coupling coefficient corresponding to the small dip frequency [f.sub.1min] will enhance the signal strength of [f.sub.1min] and reduce the signal strength corresponding to the large dip frequency [f.sub.2min]. Similarly, the influence of coupling coefficient [k.sub.2] on the signal strength of the sensor is illustrated in Figure 9. But unlike the influence of [k.sub.1], the increased [k.sub.2] not only enhances the signal strength of [f.sub.2min] significantly but also makes [DELTA][[theta].sub.1] increased obviously. It should be noted that the increase of [k.sub.1] and [k.sub.2] is usually based on the decrease of readout distance. Comprehensively analyzing the data illustrated in Figures 8(b) and 9(b), considerable signal strength of [f.sub.1min] and [f.sub.2min] can be achieved by properly reducing the value of [k.sub.1] without shortening the readout distance, that is, making [k.sub.1] less than the value of [k.sub.2].

3.1.3. Resonant Frequency Interval of the LC Tanks. The influence of resonant frequency interval [DELTA]f (equal to [f.sub.2] - [f.sub.1]) on the sensor's signal strength is illustrated in Figure 10. It should be noted that the change of [DELTA]f was realized by changing the value of [C.sub.2], that is, by only changing the value of [f.sub.2]. And the value of [R.sub.2] was changed accordingly to make the Q factor [Q.sub.2] keep constant. It is obvious that the influence of [DELTA]f on the value of [DELTA][[theta].sub.1] and [DELTA][[theta].sub.2] is similar to the decrease of [k.sub.12]. Therefore, broadening the resonant frequency interval [DELTA]f of the sensor can make the signal of [f.sub.2] strong and make [DELTA][[theta].sub.1] decreased slightly. However, larger [DELTA]f means broader sweep bandwidth which will place greater demands on the readout circuit, that is, challenging the sampling speed and accuracy of the circuit. Overall, the resonant frequency interval [DELTA]f can be properly broadened to achieve better signal strength of the sensor without shortening the readout distance and increasing the sensor size, if the bandwidth, sampling speed, and accuracy of the readout circuit can fulfill the requirements.

3.2. Influencing Factors of the Signal Crosstalk. As illustrated in Figures 4 and 10, the value of [f.sub.1min] not only depends on the corresponding resonant frequency [f.sub.1] when the coupling coefficients [k.sub.1], [k.sub.2], and [k.sub.12] keep constant, but also depends on the resonant frequency of the other resonator ([f.sub.2]), so does [f.sub.2min]. When the inductance [L.sub.1] = [L.sub.2] = L, the phase dip frequencies [f.sub.1min] and [f.sub.2min] can be derived as [7]

[f.sup.2.sub.min] L([C.sub.1] : [C.sub.2]) [+ or -] [square root of ([L.sup.2][([C.sub.1] + [C.sub.2]).sup.2] - 4[C.sub.1][C.sub.2][L.sup.2](1 - [k.sup.2.sub.12])]/8[[pi].sup.2][C.sub.1][C.sub.2][L.sup.2](1 - [k.sup.2.sub.12] (5)

It can be seen from (5) that the values of [f.sub.1min] and [f.sub.2min] only depend on the coupling coefficient [k.sub.12] and capacitances [C.sub.1] and [C.sub.2] if the inductances [L.sub.1] and [L.sub.2] keep constant. Actually, the value of [C.sub.1] and [C.sub.2] can also be expressed with abovementioned resonant frequency interval [DELTA]f. If [k.sub.12] = 0, (5) can be rewritten as

[f.sub.1min] = 1/2[pi][square root of ([LC.sub.1])] = [f.sub.1], (6a)

[f.sub.2min] = 1/2[pi][square root of ([LC.sub.2])] = [f.sub.2], (6b)

From (6a) and (6b), it can be seen that the fundamental reason for the crosstalk is the existence of mutual coupling between sensor coils. In this part, the influence of [k.sub.12] and [DELTA]f on the crosstalk strength ([DELTA][f.sub.2min_cross]) was analyzed.

3.2.1. Coupling Coefficients. As illustrated in Figure 11, the crosstalk strength [DELTA][f.sub.2min_cross] caused by the change of capacitance [C.sub.1] from 15 pF to 19 pF increased monotonously with the increase of [k.sub.12]. Specifically, when [k.sub.12] increased from 0.2 to 0.4, the crosstalk strength [DELTA][f.sub.2min] cross increased from 0.39 MHz to 1.18 MHz. Therefore, it can be concluded that reducing [k.sub.12] can significantly reduce the crosstalk between two measurement parameters.

3.2.2. Resonant Frequency Interval of the LC Tanks. As illustrated in Figure 12, the crosstalk strength [DELTA][f.sub.2min_cross] caused by the change of capacitance [C.sub.1] from 15 pF to 19 pF decreased monotonously with the increase of the resonant frequency interval [DELTA]f. Specifically, when [DELTA]f increased from 15.18 MHz to 38.48 MHz, the crosstalk strength [DELTA][f.sub.2min_cross] decreased from 0.77 MHz to 0.33 MHz. Therefore, it can be concluded that properly increasing [DELTA]f can also reduce the crosstalk between two measurement parameters.

4. Crosstalk Decoupling

4.1. Decoupling Scheme. The requirement of miniaturization needs the sensor coils overlapped as much as possible, which makes the coupling coefficient [k.sub.12] nonignorable, and it eventually makes the crosstalk nonignorable. Therefore, calibrating the sensor by directly using dip frequencies [f.sub.1min] and [f.sub.2min] will result in large measurement error. In order to avoid the error induced by the crosstalk, a decoupling scheme needs to be developed to realize accurate dual-parameter RF readout in the combinational environment. In this paper, a decoupling function as given in (7a) and (7b) will be developed to decouple the crosstalk algorithmically.

[f.sub.1] = [F.sub.1] if ([f.sub.1min], [f.sub.2min], [k.sub.12]), (7a)

[f.sub.2] = [F.sub.2] ([f.sub.1min], [f.sub.2min], [k.sub.12]). (7b)

Equations (7a) and (7b) are the function which takes [f.sub.1min], [f.sub.2min], and [k.sub.12] as the arguments and takes [f.sub.1] and [f.sub.2] as the dependent variables. The values of [f.sub.1min] and [f.sub.2min] can be readout by extracting the impedance of the readout coil, and the values of [k.sub.12] can be simulated by EM simulation software according to the layout of the sensor coils. Therefore, the resonant frequency of each LC tank can be achieved by substituting [f.sub.1min], [f.sub.2min], and [k.sub.12] into the decoupling function.

4.2. Decoupling Function Derivation. The values of dip frequencies [f.sub.1min] and [f.sub.2min] can be derived from [7]

[Z.sub.1][Z.sub.2] + [[omega].sup.2][M.sup.2.sub.12] = 0. (8)

Equation (8) can be further rewritten as

[mathematical expression not reproducible]. (9)

The left term of (9) is a complex number, and the sufficient and necessary condition for a complex number to be zero is that its real part and imaginary part should both be zero; that is,

([[omega].sup.2][k.sup.2.sub.12][L.sub.1][L.sub.2] - [[omega].sup.2][L.sub.1][L.sub.2] + [R.sub.1][R.sub.2] + [L.sub.1]/[C.sub.2] + [L.sub.2]/[C.sub.1] 1 1/[[omega].sup.2][C.sub.1][C.sub.2]) = 0. (10)

Solving (10) to obtain the expression of dip frequency [f.sub.min]

[mathematical expression not reproducible]. (11)

The term [R.sub.1][R.sub.2][C.sub.1][C.sub.2] can be omitted in (11) due to the fact that its order of magnitude is much less than that of the term [L.sub.1][C.sub.1] + [L.sub.2][C.sub.2]. Therefore, (11) can be simplified as

[mathematical expression not reproducible]. (12)

The dip frequencies [f.sub.1min] and [f.sub.2min] can be solved from (12) as

[mathematical expression not reproducible], (13a)

[mathematical expression not reproducible]. (13b)

By form variety of (3a) and (3b), it can be rewritten as

[L.sub.1][C.sub.1] = 1/4[[pi].sup.2][f.sup.2.sub.1] = [f.sub.a], (14a)

[L.sub.2][C.sub.2] = 1/4[[pi].sup.2][f.sup.2.sub.2] = [f.sub.b]. (14b)

And substituting (14a) and (14b) into (13a) and (13b), (13a) and (13b) can be simplified as

[f.sub.1min] = [square root of ([f.sub.a] + [f.sub.b] + [square root of ([([f.sub.a] + [f.sub.b]).sup.2] - 4[f.sub.a][f.sub.b](1 - [k.sup.2.sub.12])]/8[[pi].sup.2][f.sub.a][f.sub.b](1 - [k.sup.2.sub.12]))], (15a)

[f.sub.2min] = [square root of ([f.sub.a] + [f.sub.b] + [square root of ([([f.sub.a] + [f.sub.b]).sup.2] - 4[f.sub.a][f.sub.b](1 - [k.sup.2.sub.12])]/8[[pi].sup.2][f.sub.a][f.sub.b](1 - [k.sup.2.sub.12]))], (15b)

By solving (15a) and (15b), the expression of [f.sub.a] and [f.sub.b] can be derived as

[mathematical expression not reproducible], (16a)

[mathematical expression not reproducible], (16b)

Therefore, the decoupling function proposed in (7a) and (7b) can be rewritten as

[f.sub.1] = 1/2[pi] x [square root of ([f.sub.a])] (17a)

[f.sub.2] = 1/2[pi] x [square root of ([f.sub.b])]. (17b)

5. Decoupling Method Validation

To verify the validity of multiparameter signal crosstalk decoupling algorithm shown in formulas (16a) and (16b), first the effectiveness of the decoupling algorithm is verified based on theoretical values simulation. As shown in Figure 13, the corresponding LC circuit resonance frequency of the sensor can be resolved by taking [f.sub.1min] and [f.sub.2min] and coupling coefficient ([k.sub.12] = 0.5) into formulas (16a) and (16b). When one of the LC circuits of sensitive capacitance [C.sub.s1] increases from 10 pf to 20 pf, only [f.sub.1] changes in theory, while the other LC circuit resonance frequency [f.sub.2] is changeless. As shown in Figure 13, by contrasting [f.sub.1], [f.sub.2] decoupled by [f.sub.1min], [f.sub.2min], [k.sub.12] with actual [f.sub.1], [f.sub.2] in the same coordinate system, it is not difficult to find that the decoupled [f.sub.2] is greatly in accordance with the actual value, remaining almost unchanged, and the decoupled [f.sub.1] is as well in accordance with the actual [f.sub.1], presenting the same decreasing tendency. The decoupled resonant frequency is almost in accordance with the actual resonant frequency, but the decoupled [f.sub.2] slightly decreases with the decreasing of [C.sub.s1]. It is mainly due to the omission of the inductance coil resistance in the derivation of decoupling algorithm.

By contrasting the experimental and decoupled data when [k.sub.12] is 0.216 and 0.057 obtained by ADS simulation, respectively, it is seen that the decoupled [f.sub.T] variation (74.8 kHz) is half small than when [k.sub.12] is 0.057 (Figures 14 and 15). Of course, due to the omission of resistance in the derivation of decoupling algorithm, it cannot exclude the possibility for it causing decoupling precision which does not meet the actual test data of decoupling. Above all, further research needs to be conducted combining with experiments to improve the multiparameter signal crosstalk decoupling algorithm.

6. Conclusion

RF readout of dual-parameters by a novel LC-type passive sensor was theoretically analyzed. Confronting two critical problems in multiparameter RF readout, that is, problems of signal strength of the sensor and crosstalk between dip frequencies, this paper summarized the influencing factors of these two problems and characterized each factor's effect according to the numerical results based on calculating the analysis model of the sensor system. Last but not least, a decoupling function for solving the crosstalk problem was derived out. The study presented in this work provided theoretical design guidelines for the practical use of dual-parameter LC-type passive sensors.

https://doi.org/10.1155/2017/4938732

Competing Interests

The authors declare no conflict of interests.

Authors' Contributions

All works with relation to this paper have been accomplished by all authors' efforts. The idea and design of the sensor were proposed by Qiulin Tan and YanJie Guo. The experiments of the sensor were completed with the help of Guozhu Wu and Tanyong Wei. Sanmin Shen and Tao Luo designed the fabrication method of the sensor. At last, every segment related to this paper is accomplished under the guidance of Jijun Xiong. Wendong Zhang has put forward valuable suggestions for the revision of the manuscript.

Acknowledgments

This work was supported by National Natural Science Foundation of China (Grant nos. 61471324 and 51425505) and the Outstanding Young Talents Support Plan of Shanxi Province.

References

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Qiulin Tan, (1,2) Yanjie Guo, (1,2) Guozhu Wu, (1,2) Tao Luo, (1,2) Tanyong Wei, (1,2) Sanmin Shen, (1,2) Wendong Zhang, (1,2) and Jijun Xiong (1,2)

(1) Key Laboratory of Instrumentation Science & Dynamic Measurement, Ministry of Education, North University of China, Taiyuan 030051, China

(2) Science and Technology on Electronic Test & Measurement Laboratory, North University of China, Taiyuan 030051, China Correspondence should be addressed to Qiulin Tan; tanqiulin@nuc.edu.cn

Received 9 November 2016; Revised 6 February 2017; Accepted 13 February 2017; Published 13 March 2017

Academic Editor: Aiguo Song

Caption: FIGURE 1: Schematic of dual-parameters RF readout by a LC-type passive sensor

Caption: FIGURE 2: Lumped circuit model of the passive dual-parameter sensor system.

Caption: FIGURE 3: Phase versus frequency curve of the readout coil.

Caption: FIGURE 4: Schematic of the crosstalk of the dual-parameter passive sensor.

Caption: FIGURE 5: (a) Phase versus frequency curve of the readout coil when the Q factor [Q.sub.1] increased and (b) the influence of [Q.sub.1] on the value of [DELTA][[theta].sub.1] and [DELTA][[theta].sub.2].

Caption: FIGURE 6: (a) Phase versus frequency curve of the readout coil when the Q factor [Q.sub.2] increased and (b) the influence of [Q.sub.2] on the value of [DELTA][[theta].sub.1] and [DELTA][[theta].sub.2].

Caption: FIGURE 7: (a) Phase versus frequency curve of the readout coil when the coupling coefficients [k.sub.12] increased and (b) the influence of [k.sub.12] on the value of [DELTA][[theta].sub.1] and [DELTA][[theta].sub.2].

Caption: FIGURE 8: (a) Phase versus frequency curve of the readout coil when the coupling coefficients [k.sub.1] increased and (b) the influence of [k.sub.1] on the value of [DELTA][[theta].sub.1] and [DELTA][[theta].sub.2].

Caption: FIGURE 9: (a) Phase versus frequency curve of the readout coil when the coupling coefficients [k.sub.2] increased and (b) the influence of [k.sub.2] on the value of [DELTA][[theta].sub.1] and [DELTA][[theta].sub.2].

Caption: FIGURE 10: (a) Phase versus frequency curve of the readout coil when [DELTA]f increased and (b) the influence of [DELTA]f on the value of [DELTA][[theta].sub.1] and [DELTA][[theta].sub.2].

Caption: FIGURE 11: The influence of [k.sub.12] on the value of [DELTA][f.sub.2min_cross].

Caption: FIGURE 12: The influence of [DELTA]f on the value of [DELTA][f.sub.2min_cross].

Caption: FIGURE 13: Decoupling theory validation of the algorithm.

Caption: FIGURE 14: [k.sub.12] = 0.057 decoupling results compared with the experimental data.

Caption: FIGURE 15: [k.sub.12] = 0.216 decoupling results compared with the experimental data.
TABLE 1: Initial design parameters of the sensor system.

Symbol (unit)                               Value

[L.sub.1],[L.sub.2],[L.sub.a] ([mu]H)         1
[R.sub.1], [R.sub.2], [R.sub.a] ([OMEGA])     5
[C.sub.1] (pF)                               15
[C.sub.2] (pF)                                8
[k.sub.1], [k.sub.2]                         0.1
[k.sub.12]                                   0.3
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Title Annotation:Research Article
Author:Tan, Qiulin; Guo, Yanjie; Wu, Guozhu; Luo, Tao; Wei, Tanyong; Shen, Sanmin; Zhang, Wendong; Xiong, J
Publication:Modelling and Simulation in Engineering
Article Type:Report
Date:Jan 1, 2017
Words:4906
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