#### Abstract

This paper systematically studied the simultaneous measurement of two parameters by a -type passive sensor from the theoretical perspective. Based on the lumped circuit model of the typical -type passive dual-parameter sensor system, the influencing factors of the signal strength of the sensor as well as the influencing factors of signal crosstalk were both analyzed. It is found that the influencing factors of the readout signal strength of the sensor are mainly quality factors ( factors) of the tanks, coupling coefficients, and the resonant frequency interval of the two tanks. And the influencing factors of the signal crosstalk are mainly coupling coefficient between the sensor inductance coils and the resonant frequency interval of the two tanks. The specific influence behavior of corresponding influencing factors on the signal strength and crosstalk is illustrated by a series of curves from numerical results simulated by using MATLAB software. Additionally, a decoupling scheme for solving the crosstalk problem algorithmically was proposed and a corresponding function was derived out. Overall, the theoretical analysis conducted in this work can provide design guidelines for making the dual-parameter -type passive sensor useful in practical applications.

#### 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 resonators into a single sensor to realize dual-parameters 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 readout by the -type passive sensor, as well as the crosstalk decoupling scheme.

In order to provide guidelines for optimum design of the dual-parameter -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 type passive dual-parameter sensor system is illustrated in Figure 2, and the sensor is equivalent to two resonant circuits. and are the inductors of the sensor, and are the series resistances of the sensor, and and are the sensitive capacitors of the sensor. Similarly, the readout coil is equivalent to an inductor and a series resistance . 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 . When the sensor was magnetically coupled with the readout coil, there are also mutual inductance and , 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]where is the angular frequency and is equal to . And and are the coupling coefficients between the readout coil and the sensor coils. are the coupling coefficients between the sensor coils. and are the equivalent impedance of these two tanks, respectively, and they can be given asThe resonance frequency of these tanks can be written asSimilar to the readout of the single-parameter -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 tanks of the sensor, it is necessary to make the resonant frequencies and separated within the measurement range of the sensor when we design the sensor. In this paper, the inductances and are designed to be constant and the resonant frequencies changes of the sensor depended on the change of capacitances and .

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 ( = 39.56 MHz, = 61.52 MHz in this design) are not equal to the resonant frequencies ( = 41.09 MHz, = 56.27 MHz) of the sensor due to the mutual coupling . Specifically, the mutual coupling between the sensor coils makes the phase dip frequencies apart compared to the resonant frequencies; that is, the dip frequency is smaller than the corresponding resonant frequency and is larger than .

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 deceased monotonously when the capacitance increased from 15 pF to 19 pF (other parameters keep constant). It should be noted that the change of results in the drift of the dip frequency , and this phenomenon is a kind of crosstalk which is originally caused by the mutual coupling () between the sensor coils. The crosstalk strength can be indicated by the drift value . Usually, the magnitude of the phase dip ( and 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 and . 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 corresponding to the resonant point of a passive sensor will decrease at elevated temperatures due to the degraded factor of the resonator, which will make tracking the frequencies and 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 and 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 and need to be analyzed.

###### 3.1.1. Quality Factor

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

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###### 3.1.2. Coupling Coefficients

In the dual-parameter passive sensor system, there are three coupling coefficients , , and . As illustrated in Figure 7(a), the increase of not only made the phase difference and and changed, but also made the obvious drift of and . Specifically, decreased and increased when increased from 0.2 to 0.4 (other parameters keep constant). It can be seen from Figure 7(b) that the phase difference increased from 39.9° to 43.8° when increased from 0.2 to 0.4, but decreased from 14.6° to 4.6°. Overall, increasing the coupling coefficient makes phase difference corresponding to the small frequency point increased slightly and makes the phase difference corresponding to decreased obviously. In the dual-parameter passive sensor system, the signal strength corresponding to the larger frequency was usually suppressed due to mutual coupling between the sensor coils. Therefore, considerable sensor strength can also be achieved by reducing , 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 if other restrictions permit.

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It is obvious from Figure 8 that change of the coupling coefficient has significant influence on the value of and . When increased from 0.08 to 0.12, increased from 31.3° to 54.6° and decreased from 11.9° to 5.1° accordingly. Therefore, the increase of the coupling coefficient corresponding to the small dip frequency will enhance the signal strength of and reduce the signal strength corresponding to the large dip frequency . Similarly, the influence of coupling coefficient on the signal strength of the sensor is illustrated in Figure 9. But unlike the influence of , the increased not only enhances the signal strength of significantly but also makes Δ*θ*_{1} increased obviously. It should be noted that the increase of and 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 and can be achieved by properly reducing the value of without shortening the readout distance, that is, making less than the value of .

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###### 3.1.3. Resonant Frequency Interval of the Tanks

The influence of resonant frequency interval (equal to ) on the sensor’s signal strength is illustrated in Figure 10. It should be noted that the change of was realized by changing the value of , that is, by only changing the value of . And the value of was changed accordingly to make the factor keep constant. It is obvious that the influence of on the value of and is similar to the decrease of . Therefore, broadening the resonant frequency interval of the sensor can make the signal of strong and make decreased slightly. However, larger 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 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.

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##### 3.2. Influencing Factors of the Signal Crosstalk

As illustrated in Figures 4 and 10, the value of not only depends on the corresponding resonant frequency when the coupling coefficients , , and keep constant, but also depends on the resonant frequency of the other resonator (), so does . When the inductance , the phase dip frequencies and can be derived as [7]It can be seen from (5) that the values of and only depend on the coupling coefficient and capacitances and if the inductances and keep constant. Actually, the value of and can also be expressed with above-mentioned resonant frequency interval . If , (5) can be rewritten asFrom (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 and on the crosstalk strength () was analyzed.

###### 3.2.1. Coupling Coefficients

As illustrated in Figure 11, the crosstalk strength caused by the change of capacitance from 15 pF to 19 pF increased monotonously with the increase of . Specifically, when increased from 0.2 to 0.4, the crosstalk strength increased from 0.39 MHz to 1.18 MHz. Therefore, it can be concluded that reducing can significantly reduce the crosstalk between two measurement parameters.

###### 3.2.2. Resonant Frequency Interval of the Tanks

As illustrated in Figure 12, the crosstalk strength caused by the change of capacitance from 15 pF to 19 pF decreased monotonously with the increase of the resonant frequency interval . Specifically, when increased from 15.18 MHz to 38.48 MHz, the crosstalk strength decreased from 0.77 MHz to 0.33 MHz. Therefore, it can be concluded that properly increasing 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 nonignorable, and it eventually makes the crosstalk nonignorable. Therefore, calibrating the sensor by directly using dip frequencies and 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 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.Equations (7a) and (7b) are the function which takes , , and as the arguments and takes and as the dependent variables. The values of and can be readout by extracting the impedance of the readout coil, and the values of can be simulated by EM simulation software according to the layout of the sensor coils. Therefore, the resonant frequency of each tank can be achieved by substituting , , and into the decoupling function.

##### 4.2. Decoupling Function Derivation

The values of dip frequencies and can be derived from [7]Equation (8) can be further rewritten asThe 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,Solving (10) to obtain the expression of dip frequency ,The term can be omitted in (11) due to the fact that its order of magnitude is much less than that of the term . Therefore, (11) can be simplified asThe dip frequencies and can be solved from (12) asBy form variety of (3a) and (3b), it can be rewritten asAnd substituting (14a) and (14b) into (13a) and (13b), (13a) and (13b) can be simplified asBy solving (15a) and (15b), the expression of and can be derived asTherefore, the decoupling function proposed in (7a) and (7b) can be rewritten as

#### 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 circuit resonance frequency of the sensor can be resolved by taking and and coupling coefficient () into formulas (16a) and (16b). When one of the circuits of sensitive capacitance increases from 10 pf to 20 pf, only changes in theory, while the other circuit resonance frequency is changeless. As shown in Figure 13, by contrasting , decoupled by , , with actual , in the same coordinate system, it is not difficult to find that the decoupled is greatly in accordance with the actual value, remaining almost unchanged, and the decoupled is as well in accordance with the actual , presenting the same decreasing tendency. The decoupled resonant frequency is almost in accordance with the actual resonant frequency, but the decoupled slightly decreases with the decreasing of . 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 is 0.216 and 0.057 obtained by ADS simulation, respectively, it is seen that the decoupled variation (74.8 kHz) is half small than when 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.

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#### 6. Conclusion

readout of dual-parameters by a novel -type passive sensor was theoretically analyzed. Confronting two critical problems in multiparameter 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 -type passive sensors.

#### 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.