# Systematic Theoretical Analysis of Dual-ParametersRF Readout by a Novel LC-Type Passive Sensor

Authors: Qiulin Tan; Yanjie Guo; Guozhu Wu; Tao Luo; Tanyong Wei; Sanmin Shen; Wendong Zhang; Jijun Xiong
Journal: Modelling and Simulation in Engineering
 (2017)

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## Abstract

This paper systematically studied the simultaneous measurement of two parameters by aLC-type passive sensor from the theoretical perspective. Based on the lumped circuit model of the typical LC-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 RF readout signal strength of the sensor are mainly quality factors (Q factors) of the LC tanks, coupling coefficients, and the resonant frequency interval of the two LC 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 LC 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 LC-type passive sensor useful in practical applications.


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## Body

## 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 twoLC 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.Figure 1
Schematic of dual-parametersRF readout by a LC-type passive sensor.In order to provide guidelines for optimum design of the dual-parameterLC-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 theLC type passive dual-parameter sensor system is illustrated in Figure 2, and the sensor is equivalent to two LC resonant circuits. L1 and L2 are the inductors of the sensor, R1 and R2 are the series resistances of the sensor, and C1 and C2 are the sensitive capacitors of the sensor. Similarly, the readout coil is equivalent to an inductor La and a series resistance Ra. 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 M12. When the sensor was magnetically coupled with the readout coil, there are also mutual inductance M1 and M2, which corresponds to the coupling between the readout coil and the sensor inductors.Figure 2
Lumped circuit model of the passive dual-parameter sensor system.The input impedance of the readout coil can be given by [7](1)Zin=Ra+jωLa+jωM12-ω2M2M12-jωM1Z2Z1Z2+ω2M122+jωM2-ω2M1M12-jωM2Z1Z1Z2+ω2M122=Ra+jωLa+jωk1L1La-ω2k2k12L2LaL1-jωk1LaL1Z2Z1Z2+ω2k122L1L2+jωk2L2La-ω2k1k12L1LaL2-jωk2LaL2Z1Z1Z2+ω2k122L1L2,where ω is the angular frequency and is equal to 2πf. And k1 and k2 are the coupling coefficients between the readout coil and the sensor coils. k12 are the coupling coefficients between the sensor coils. Z1 and Z2 are the equivalent impedance of these two LC tanks, respectively, and they can be given as(2a)Z1=R1+jwL1+1jwC1,(2b)Z2=R2+jωL2+1jωC2.The resonance frequency of these LC tanks can be written as(3a)f1=12πL1C1,(3b)f2=12πL2C2.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 f1 and f2 separated within the measurement range of the sensor when we design the sensor. In this paper, the inductances L1 and L2 are designed to be constant and the resonant frequencies changes of the sensor depended on the change of capacitances C1 and C2.Herein, by substituting the parameters in Table1 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 (f1min = 39.56 MHz, f2min = 61.52 MHz in this design) are not equal to the resonant frequencies (f1 = 41.09 MHz, f2 = 56.27 MHz) of the sensor due to the mutual coupling k12. Specifically, the mutual coupling between the sensor coils makes the phase dip frequencies apart compared to the resonant frequencies; that is, the dip frequency f1min is smaller than the corresponding resonant frequency f1 and f2min is larger than f2.Table 1
Initial design parameters of the sensor system.
Symbol (unit)
Value
L
1, L2, La (μH)
1
R
1, R2, Ra (Ω)
5
C
1 (pF)
15
C
2 (pF)
8
k
1, k2
0.1
k
12
0.3Figure 3
Phase versus frequency curve of the readout coil.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 Figure4, the dip frequency f1min deceased monotonously when the capacitance C1 increased from 15 pF to 19 pF (other parameters keep constant). It should be noted that the change of f1 results in the drift of the dip frequency f2min, and this phenomenon is a kind of crosstalk which is originally caused by the mutual coupling (k12) between the sensor coils. The crosstalk strength can be indicated by the drift value Δf2min_cross. Usually, the magnitude of the phase dip (Δθ1 and Δθ2 in Figure 2) should be large enough to make the readout of two phase dip frequencies possible within the measurement range.Figure 4
Schematic of the crosstalk of the dual-parameter passive sensor.


## 3. Numerical Simulation and Analysis

### 3.1. Influencing Factors of the Sensor’s Signal Strength

As illustrated in Figure3, simultaneous dual-parameter measurement can be realized by tracking the dip frequencies f1min and f2min. 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 Δθ 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 f1min and f2min 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 Δθ1 and Δθ2 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 Δθ1 and Δθ2 need to be analyzed.


#### 3.1.1. Quality Factor

As for aLC resonator, its Q factor can be given as(4)Q=ω0LsRs=12πRsLsCs.It is obvious from (4) that the Q factor of the LC resonator mainly depends on its equivalent series resistance. Therefore, the resistance R1 and resistance R2 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 Δθ1 increased obviously with the increase of Q factor Q1, and Δθ2 also showed a slight increase when Q1 increased. It can be seen from Figure 6 that the increase of Q2 resulted in an obvious increase of Δθ2 and a slight increase of Δθ1. 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.Figure 5
(a) Phase versus frequency curve of the readout coil when theQ factor Q1 increased and (b) the influence of Q1 on the value of Δθ1 and Δθ2.
(a)
(b)Figure 6
(a) Phase versus frequency curve of the readout coil when theQ factor Q2 increased and (b) the influence of Q2 on the value of Δθ1 and Δθ2.
(a)
(b)


#### 3.1.2. Coupling Coefficients

In the dual-parameter passive sensor system, there are three coupling coefficientsk1, k2, and k12. As illustrated in Figure 7(a), the increase of k12 not only made the phase difference Δθ1 and Δθ2 and changed, but also made the obvious drift of f1min and f2min. Specifically, f1min decreased and f2min increased when k12 increased from 0.2 to 0.4 (other parameters keep constant). It can be seen from Figure 7(b) that the phase difference Δθ1 increased from 39.9° to 43.8° when k12 increased from 0.2 to 0.4, but Δθ2 decreased from 14.6° to 4.6°. Overall, increasing the coupling coefficient k12 makes phase difference corresponding to the small frequency point f1min increased slightly and makes the phase difference corresponding to f2min decreased obviously. In the dual-parameter passive sensor system, the signal strength corresponding to the larger frequency f2min was usually suppressed due to mutual coupling between the sensor coils. Therefore, considerable sensor strength can also be achieved by reducing k12, 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 k12 if other restrictions permit.Figure 7
(a) Phase versus frequency curve of the readout coil when the coupling coefficientsk12 increased and (b) the influence of k12 on the value of Δθ1 and Δθ2.
(a)
(b)It is obvious from Figure8 that change of the coupling coefficient k1 has significant influence on the value of Δθ1 and Δθ2. When k1 increased from 0.08 to 0.12, Δθ1 increased from 31.3° to 54.6° and Δθ2 decreased from 11.9° to 5.1° accordingly. Therefore, the increase of the coupling coefficient corresponding to the small dip frequency f1min will enhance the signal strength of f1min and reduce the signal strength corresponding to the large dip frequency f2min. Similarly, the influence of coupling coefficient k2 on the signal strength of the sensor is illustrated in Figure 9. But unlike the influence of k1, the increased k2 not only enhances the signal strength of f2min significantly but also makes Δθ1 increased obviously. It should be noted that the increase of k1 and k2 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 f1min and f2min can be achieved by properly reducing the value of k1 without shortening the readout distance, that is, making k1 less than the value of k2.Figure 8
(a) Phase versus frequency curve of the readout coil when the coupling coefficientsk1 increased and (b) the influence of k1 on the value of Δθ1 and Δθ2.
(a)
(b)Figure 9
(a) Phase versus frequency curve of the readout coil when the coupling coefficientsk2 increased and (b) the influence of k2 on the value of Δθ1 and Δθ2.
(a)
(b)


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

The influence of resonant frequency intervalΔf (equal to f2-f1) on the sensor’s signal strength is illustrated in Figure 10. It should be noted that the change of Δf was realized by changing the value of C2, that is, by only changing the value of f2. And the value of R2 was changed accordingly to make the Q factor Q2 keep constant. It is obvious that the influence of Δf on the value of Δθ1 and Δθ2 is similar to the decrease of k12. Therefore, broadening the resonant frequency interval Δf of the sensor can make the signal of f2 strong and make Δθ1 decreased slightly. However, larger Δf 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 Δf 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.Figure 10
(a) Phase versus frequency curve of the readout coil whenΔf increased and (b) the influence of Δf on the value of Δθ1 and Δθ2.
(a)
(b)


### 3.2. Influencing Factors of the Signal Crosstalk

As illustrated in Figures4 and 10, the value of f1min not only depends on the corresponding resonant frequency f1 when the coupling coefficients k1, k2, and k12 keep constant, but also depends on the resonant frequency of the other resonator (f2), so does f2min. When the inductance L1=L2=L, the phase dip frequencies f1min and f2min can be derived as [7](5)fmin2=LC1+C2±L2C1+C22-4C1C2L21-k1228π2C1C2L21-k122.It can be seen from (5) that the values of f1min and f2min only depend on the coupling coefficient k12 and capacitances C1 and C2 if the inductances L1 and L2 keep constant. Actually, the value of C1 and C2 can also be expressed with above-mentioned resonant frequency interval Δf. If k12=0, (5) can be rewritten as(6a)f1min=12πLC1=f1,(6b)f2min=12πLC2=f2.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 k12 and Δf on the crosstalk strength (Δf2min_cross) was analyzed.


#### 3.2.1. Coupling Coefficients

As illustrated in Figure11, the crosstalk strength Δf2min_cross caused by the change of capacitance C1 from 15 pF to 19 pF increased monotonously with the increase of k12. Specifically, when k12 increased from 0.2 to 0.4, the crosstalk strength Δf2min_cross increased from 0.39 MHz to 1.18 MHz. Therefore, it can be concluded that reducing k12 can significantly reduce the crosstalk between two measurement parameters.Figure 11
The influence ofk12 on the value ofΔf2min_cross.


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

As illustrated in Figure12, the crosstalk strength Δf2min_cross caused by the change of capacitance C1 from 15 pF to 19 pF decreased monotonously with the increase of the resonant frequency interval Δf. Specifically, when Δf increased from 15.18 MHz to 38.48 MHz, the crosstalk strength Δf2min_cross decreased from 0.77 MHz to 0.33 MHz. Therefore, it can be concluded that properly increasing Δf can also reduce the crosstalk between two measurement parameters.Figure 12
The influence ofΔf on the value of Δf2min_cross.


## 3.1. Influencing Factors of the Sensor’s Signal Strength

As illustrated in Figure3, simultaneous dual-parameter measurement can be realized by tracking the dip frequencies f1min and f2min. 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 Δθ 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 f1min and f2min 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 Δθ1 and Δθ2 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 Δθ1 and Δθ2 need to be analyzed.


### 3.1.1. Quality Factor

As for aLC resonator, its Q factor can be given as(4)Q=ω0LsRs=12πRsLsCs.It is obvious from (4) that the Q factor of the LC resonator mainly depends on its equivalent series resistance. Therefore, the resistance R1 and resistance R2 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 Δθ1 increased obviously with the increase of Q factor Q1, and Δθ2 also showed a slight increase when Q1 increased. It can be seen from Figure 6 that the increase of Q2 resulted in an obvious increase of Δθ2 and a slight increase of Δθ1. 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.Figure 5
(a) Phase versus frequency curve of the readout coil when theQ factor Q1 increased and (b) the influence of Q1 on the value of Δθ1 and Δθ2.
(a)
(b)Figure 6
(a) Phase versus frequency curve of the readout coil when theQ factor Q2 increased and (b) the influence of Q2 on the value of Δθ1 and Δθ2.
(a)
(b)


### 3.1.2. Coupling Coefficients

In the dual-parameter passive sensor system, there are three coupling coefficientsk1, k2, and k12. As illustrated in Figure 7(a), the increase of k12 not only made the phase difference Δθ1 and Δθ2 and changed, but also made the obvious drift of f1min and f2min. Specifically, f1min decreased and f2min increased when k12 increased from 0.2 to 0.4 (other parameters keep constant). It can be seen from Figure 7(b) that the phase difference Δθ1 increased from 39.9° to 43.8° when k12 increased from 0.2 to 0.4, but Δθ2 decreased from 14.6° to 4.6°. Overall, increasing the coupling coefficient k12 makes phase difference corresponding to the small frequency point f1min increased slightly and makes the phase difference corresponding to f2min decreased obviously. In the dual-parameter passive sensor system, the signal strength corresponding to the larger frequency f2min was usually suppressed due to mutual coupling between the sensor coils. Therefore, considerable sensor strength can also be achieved by reducing k12, 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 k12 if other restrictions permit.Figure 7
(a) Phase versus frequency curve of the readout coil when the coupling coefficientsk12 increased and (b) the influence of k12 on the value of Δθ1 and Δθ2.
(a)
(b)It is obvious from Figure8 that change of the coupling coefficient k1 has significant influence on the value of Δθ1 and Δθ2. When k1 increased from 0.08 to 0.12, Δθ1 increased from 31.3° to 54.6° and Δθ2 decreased from 11.9° to 5.1° accordingly. Therefore, the increase of the coupling coefficient corresponding to the small dip frequency f1min will enhance the signal strength of f1min and reduce the signal strength corresponding to the large dip frequency f2min. Similarly, the influence of coupling coefficient k2 on the signal strength of the sensor is illustrated in Figure 9. But unlike the influence of k1, the increased k2 not only enhances the signal strength of f2min significantly but also makes Δθ1 increased obviously. It should be noted that the increase of k1 and k2 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 f1min and f2min can be achieved by properly reducing the value of k1 without shortening the readout distance, that is, making k1 less than the value of k2.Figure 8
(a) Phase versus frequency curve of the readout coil when the coupling coefficientsk1 increased and (b) the influence of k1 on the value of Δθ1 and Δθ2.
(a)
(b)Figure 9
(a) Phase versus frequency curve of the readout coil when the coupling coefficientsk2 increased and (b) the influence of k2 on the value of Δθ1 and Δθ2.
(a)
(b)


### 3.1.3. Resonant Frequency Interval of theLC Tanks

The influence of resonant frequency intervalΔf (equal to f2-f1) on the sensor’s signal strength is illustrated in Figure 10. It should be noted that the change of Δf was realized by changing the value of C2, that is, by only changing the value of f2. And the value of R2 was changed accordingly to make the Q factor Q2 keep constant. It is obvious that the influence of Δf on the value of Δθ1 and Δθ2 is similar to the decrease of k12. Therefore, broadening the resonant frequency interval Δf of the sensor can make the signal of f2 strong and make Δθ1 decreased slightly. However, larger Δf 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 Δf 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.Figure 10
(a) Phase versus frequency curve of the readout coil whenΔf increased and (b) the influence of Δf on the value of Δθ1 and Δθ2.
(a)
(b)


## 3.1.1. Quality Factor

As for aLC resonator, its Q factor can be given as(4)Q=ω0LsRs=12πRsLsCs.It is obvious from (4) that the Q factor of the LC resonator mainly depends on its equivalent series resistance. Therefore, the resistance R1 and resistance R2 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 Δθ1 increased obviously with the increase of Q factor Q1, and Δθ2 also showed a slight increase when Q1 increased. It can be seen from Figure 6 that the increase of Q2 resulted in an obvious increase of Δθ2 and a slight increase of Δθ1. 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.Figure 5
(a) Phase versus frequency curve of the readout coil when theQ factor Q1 increased and (b) the influence of Q1 on the value of Δθ1 and Δθ2.
(a)
(b)Figure 6
(a) Phase versus frequency curve of the readout coil when theQ factor Q2 increased and (b) the influence of Q2 on the value of Δθ1 and Δθ2.
(a)
(b)


## 3.1.2. Coupling Coefficients

In the dual-parameter passive sensor system, there are three coupling coefficientsk1, k2, and k12. As illustrated in Figure 7(a), the increase of k12 not only made the phase difference Δθ1 and Δθ2 and changed, but also made the obvious drift of f1min and f2min. Specifically, f1min decreased and f2min increased when k12 increased from 0.2 to 0.4 (other parameters keep constant). It can be seen from Figure 7(b) that the phase difference Δθ1 increased from 39.9° to 43.8° when k12 increased from 0.2 to 0.4, but Δθ2 decreased from 14.6° to 4.6°. Overall, increasing the coupling coefficient k12 makes phase difference corresponding to the small frequency point f1min increased slightly and makes the phase difference corresponding to f2min decreased obviously. In the dual-parameter passive sensor system, the signal strength corresponding to the larger frequency f2min was usually suppressed due to mutual coupling between the sensor coils. Therefore, considerable sensor strength can also be achieved by reducing k12, 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 k12 if other restrictions permit.Figure 7
(a) Phase versus frequency curve of the readout coil when the coupling coefficientsk12 increased and (b) the influence of k12 on the value of Δθ1 and Δθ2.
(a)
(b)It is obvious from Figure8 that change of the coupling coefficient k1 has significant influence on the value of Δθ1 and Δθ2. When k1 increased from 0.08 to 0.12, Δθ1 increased from 31.3° to 54.6° and Δθ2 decreased from 11.9° to 5.1° accordingly. Therefore, the increase of the coupling coefficient corresponding to the small dip frequency f1min will enhance the signal strength of f1min and reduce the signal strength corresponding to the large dip frequency f2min. Similarly, the influence of coupling coefficient k2 on the signal strength of the sensor is illustrated in Figure 9. But unlike the influence of k1, the increased k2 not only enhances the signal strength of f2min significantly but also makes Δθ1 increased obviously. It should be noted that the increase of k1 and k2 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 f1min and f2min can be achieved by properly reducing the value of k1 without shortening the readout distance, that is, making k1 less than the value of k2.Figure 8
(a) Phase versus frequency curve of the readout coil when the coupling coefficientsk1 increased and (b) the influence of k1 on the value of Δθ1 and Δθ2.
(a)
(b)Figure 9
(a) Phase versus frequency curve of the readout coil when the coupling coefficientsk2 increased and (b) the influence of k2 on the value of Δθ1 and Δθ2.
(a)
(b)


## 3.1.3. Resonant Frequency Interval of theLC Tanks

The influence of resonant frequency intervalΔf (equal to f2-f1) on the sensor’s signal strength is illustrated in Figure 10. It should be noted that the change of Δf was realized by changing the value of C2, that is, by only changing the value of f2. And the value of R2 was changed accordingly to make the Q factor Q2 keep constant. It is obvious that the influence of Δf on the value of Δθ1 and Δθ2 is similar to the decrease of k12. Therefore, broadening the resonant frequency interval Δf of the sensor can make the signal of f2 strong and make Δθ1 decreased slightly. However, larger Δf 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 Δf 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.Figure 10
(a) Phase versus frequency curve of the readout coil whenΔf increased and (b) the influence of Δf on the value of Δθ1 and Δθ2.
(a)
(b)


## 3.2. Influencing Factors of the Signal Crosstalk

As illustrated in Figures4 and 10, the value of f1min not only depends on the corresponding resonant frequency f1 when the coupling coefficients k1, k2, and k12 keep constant, but also depends on the resonant frequency of the other resonator (f2), so does f2min. When the inductance L1=L2=L, the phase dip frequencies f1min and f2min can be derived as [7](5)fmin2=LC1+C2±L2C1+C22-4C1C2L21-k1228π2C1C2L21-k122.It can be seen from (5) that the values of f1min and f2min only depend on the coupling coefficient k12 and capacitances C1 and C2 if the inductances L1 and L2 keep constant. Actually, the value of C1 and C2 can also be expressed with above-mentioned resonant frequency interval Δf. If k12=0, (5) can be rewritten as(6a)f1min=12πLC1=f1,(6b)f2min=12πLC2=f2.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 k12 and Δf on the crosstalk strength (Δf2min_cross) was analyzed.


### 3.2.1. Coupling Coefficients

As illustrated in Figure11, the crosstalk strength Δf2min_cross caused by the change of capacitance C1 from 15 pF to 19 pF increased monotonously with the increase of k12. Specifically, when k12 increased from 0.2 to 0.4, the crosstalk strength Δf2min_cross increased from 0.39 MHz to 1.18 MHz. Therefore, it can be concluded that reducing k12 can significantly reduce the crosstalk between two measurement parameters.Figure 11
The influence ofk12 on the value ofΔf2min_cross.


### 3.2.2. Resonant Frequency Interval of theLC Tanks

As illustrated in Figure12, the crosstalk strength Δf2min_cross caused by the change of capacitance C1 from 15 pF to 19 pF decreased monotonously with the increase of the resonant frequency interval Δf. Specifically, when Δf increased from 15.18 MHz to 38.48 MHz, the crosstalk strength Δf2min_cross decreased from 0.77 MHz to 0.33 MHz. Therefore, it can be concluded that properly increasing Δf can also reduce the crosstalk between two measurement parameters.Figure 12
The influence ofΔf on the value of Δf2min_cross.


## 3.2.1. Coupling Coefficients

As illustrated in Figure11, the crosstalk strength Δf2min_cross caused by the change of capacitance C1 from 15 pF to 19 pF increased monotonously with the increase of k12. Specifically, when k12 increased from 0.2 to 0.4, the crosstalk strength Δf2min_cross increased from 0.39 MHz to 1.18 MHz. Therefore, it can be concluded that reducing k12 can significantly reduce the crosstalk between two measurement parameters.Figure 11
The influence ofk12 on the value ofΔf2min_cross.


## 3.2.2. Resonant Frequency Interval of theLC Tanks

As illustrated in Figure12, the crosstalk strength Δf2min_cross caused by the change of capacitance C1 from 15 pF to 19 pF decreased monotonously with the increase of the resonant frequency interval Δf. Specifically, when Δf increased from 15.18 MHz to 38.48 MHz, the crosstalk strength Δf2min_cross decreased from 0.77 MHz to 0.33 MHz. Therefore, it can be concluded that properly increasing Δf can also reduce the crosstalk between two measurement parameters.Figure 12
The influence ofΔf on the value of Δf2min_cross.


## 4. Crosstalk Decoupling

### 4.1. Decoupling Scheme

The requirement of miniaturization needs the sensor coils overlapped as much as possible, which makes the coupling coefficientk12 nonignorable, and it eventually makes the crosstalk nonignorable. Therefore, calibrating the sensor by directly using dip frequencies f1min and f2min 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.(7a)f1=F1f1min,f2min,k12,(7b)f2=F2f1min,f2min,k12.Equations (7a) and (7b) are the function which takes f1min, f2min, and k12 as the arguments and takes f1 and f2 as the dependent variables. The values of f1min and f2min can be readout by extracting the impedance of the readout coil, and the values of k12 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 f1min, f2min, and k12 into the decoupling function.


### 4.2. Decoupling Function Derivation

The values of dip frequenciesf1min and f2min can be derived from [7](8)Z1Z2+ω2M122=0.Equation (8) can be further rewritten as(9)R1+jωL1+jωC1-1·R2+jωL2+jωC2-1+ω2k122L1L2=0⟹R1R2+jωL2R1+R1jωC2+jωL1R2-ω2L1L2+L1C2+R2jωC1+L2C1-1ω2C1C2+ω2k122L1L2=0⟹ω2k122L1L2-ω2L1L2+R1R2+L1C2+L2C1-1ω2C1C2︸real part+jωL2R1+R1jωCs2+jωL1R2+R2jωCs1=0.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,(10)ω2k122L1L2-ω2L1L2+R1R2+L1C2+L2C1-1ω2C1C2=0.Solving (10) to obtain the expression of dip frequency fmin,(11)ω2=2πfmin2=L1C1+L2C2+R1R2C1C22L1L2C1C21-k122±L1C1+L2C2+R1R2C1C22-4L1L2C1C21-k1222L1L2C1C21-k122.The term R1R2C1C2 can be omitted in (11) due to the fact that its order of magnitude is much less than that of the term L1C1+L2C2. Therefore, (11) can be simplified as(12)ω2=2πfmin2=L1C1+L2C2±L1C1+L2C22-4L1L2C1C21-k1222L1L2C1C21-k122.The dip frequencies f1min and f2min can be solved from (12) as(13a)f1min=L1C1+L2C2+L1C1+L2C22-4L1L2C1C21-k1228π2L1L2C1C21-k122,(13b)f2min=L1C1+L2C2-L1C1+L2C22-4L1L2C1C21-k1228π2L1L2C1C21-k122.By form variety of (3a) and (3b), it can be rewritten as(14a)L1C1=14π2f12=fa,(14b)L2C2=14π2f22=fb.And substituting (14a) and (14b) into (13a) and (13b), (13a) and (13b) can be simplified as(15a)f1min=fa+fb+fa+fb2-4fafb1-k1228π2fafb1-k122,(15b)f2min=fa+fb-fa+fb2-4fafb1-k1228π2fafb1-k122.By solving (15a) and (15b), the expression of fa and fb can be derived as(16a)fb=f1min2+f2min2+f1min2+f2min22-4f1min2·f2min2·1/1-k1228π2f1min2·f2min2,(16b)fa=f1min2+f2min2-f1min2+f2min22-4f1min2·f2min2·1/1-k1228π2f1min2·f2min2.Therefore, the decoupling function proposed in (7a) and (7b) can be rewritten as(17a)f1=12π·fa,(17b)f2=12π·fb.


## 4.1. Decoupling Scheme

The requirement of miniaturization needs the sensor coils overlapped as much as possible, which makes the coupling coefficientk12 nonignorable, and it eventually makes the crosstalk nonignorable. Therefore, calibrating the sensor by directly using dip frequencies f1min and f2min 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.(7a)f1=F1f1min,f2min,k12,(7b)f2=F2f1min,f2min,k12.Equations (7a) and (7b) are the function which takes f1min, f2min, and k12 as the arguments and takes f1 and f2 as the dependent variables. The values of f1min and f2min can be readout by extracting the impedance of the readout coil, and the values of k12 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 f1min, f2min, and k12 into the decoupling function.


## 4.2. Decoupling Function Derivation

The values of dip frequenciesf1min and f2min can be derived from [7](8)Z1Z2+ω2M122=0.Equation (8) can be further rewritten as(9)R1+jωL1+jωC1-1·R2+jωL2+jωC2-1+ω2k122L1L2=0⟹R1R2+jωL2R1+R1jωC2+jωL1R2-ω2L1L2+L1C2+R2jωC1+L2C1-1ω2C1C2+ω2k122L1L2=0⟹ω2k122L1L2-ω2L1L2+R1R2+L1C2+L2C1-1ω2C1C2︸real part+jωL2R1+R1jωCs2+jωL1R2+R2jωCs1=0.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,(10)ω2k122L1L2-ω2L1L2+R1R2+L1C2+L2C1-1ω2C1C2=0.Solving (10) to obtain the expression of dip frequency fmin,(11)ω2=2πfmin2=L1C1+L2C2+R1R2C1C22L1L2C1C21-k122±L1C1+L2C2+R1R2C1C22-4L1L2C1C21-k1222L1L2C1C21-k122.The term R1R2C1C2 can be omitted in (11) due to the fact that its order of magnitude is much less than that of the term L1C1+L2C2. Therefore, (11) can be simplified as(12)ω2=2πfmin2=L1C1+L2C2±L1C1+L2C22-4L1L2C1C21-k1222L1L2C1C21-k122.The dip frequencies f1min and f2min can be solved from (12) as(13a)f1min=L1C1+L2C2+L1C1+L2C22-4L1L2C1C21-k1228π2L1L2C1C21-k122,(13b)f2min=L1C1+L2C2-L1C1+L2C22-4L1L2C1C21-k1228π2L1L2C1C21-k122.By form variety of (3a) and (3b), it can be rewritten as(14a)L1C1=14π2f12=fa,(14b)L2C2=14π2f22=fb.And substituting (14a) and (14b) into (13a) and (13b), (13a) and (13b) can be simplified as(15a)f1min=fa+fb+fa+fb2-4fafb1-k1228π2fafb1-k122,(15b)f2min=fa+fb-fa+fb2-4fafb1-k1228π2fafb1-k122.By solving (15a) and (15b), the expression of fa and fb can be derived as(16a)fb=f1min2+f2min2+f1min2+f2min22-4f1min2·f2min2·1/1-k1228π2f1min2·f2min2,(16b)fa=f1min2+f2min2-f1min2+f2min22-4f1min2·f2min2·1/1-k1228π2f1min2·f2min2.Therefore, the decoupling function proposed in (7a) and (7b) can be rewritten as(17a)f1=12π·fa,(17b)f2=12π·fb.


## 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 f1min and f2min and coupling coefficient (k12=0.5) into formulas (16a) and (16b). When one of the LC circuits of sensitive capacitance Cs1 increases from 10 pf to 20 pf, only f1 changes in theory, while the other LC circuit resonance frequency f2 is changeless. As shown in Figure 13, by contrasting f1, f2 decoupled by f1min, f2min, k12 with actual f1, f2 in the same coordinate system, it is not difficult to find that the decoupled f2 is greatly in accordance with the actual value, remaining almost unchanged, and the decoupled f1 is as well in accordance with the actual f1, presenting the same decreasing tendency. The decoupled resonant frequency is almost in accordance with the actual resonant frequency, but the decoupled f2 slightly decreases with the decreasing of Cs1. It is mainly due to the omission of the inductance coil resistance in the derivation of decoupling algorithm.Figure 13
Decoupling theory validation of the algorithm.By contrasting the experimental and decoupled data whenk12 is 0.216 and 0.057 obtained by ADS simulation, respectively, it is seen that the decoupled fT variation (74.8 kHz) is half small than when k12 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.Figure 14
k
12
=
0.057 decoupling results compared with the experimental data.
(a)
(b)Figure 15
k
12
=
0.216 decoupling results compared with the experimental data.
(a)
(b)


## 6. Conclusion

R
F 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.



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*Source: 4938732-2017-03-13.xml*