# Coupling Influence on Signal Readout of a Dual-Parameter LC Resonant System

Authors: Jijun Xiong; Tanyong Wei; Tao Luo; Qiulin Tan; Chenyang Xue; Jun Liu; Wendong Zhang
Journal: Advances in Mathematical Physics
 (2015)

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## Abstract

Dual-parameter inductive-capacitive (LC) resonant sensor is gradually becoming the measurement trend in complex harsh environments; however, the coupling between inductors greatly affects the readout signal, which becomes very difficult to resolve by means of simple mathematical tools. By changing the values of specific variables in a MATLAB code, the influence of coupling between coils on the readout signal is analyzed. Our preliminary conclusions underline that changing the coupling to antenna greatly affects the readout signal, but it simultaneously influences the other signal. Whenf01=f02, it is better to broaden the difference between the two coupling coefficients k1 and k2. On the other side, when f01 is smaller than f02, it is better to decrease the coupling between sensor inductors k12, in order to obtain two readout signals averaged in strength. Finally, a test system including a discrete capacitor soldered to a printed circuit board (PCB) based planar spiral coil is built, and the readout signals under different relative inductors positions are analyzed. All experimental results are in good agreement with the results of the MATLAB simulation.


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## Body

## 1. Introduction

A dual-parameter test is urgently needed in the fields of aviation, mining, and automation industries. In these industries, a dual-parameter test is required for specific applications such as blade temperature and pressure monitoring in turbine engines [1–3], temperature and vibration monitoring in key components of coal mining machines [4], and monitoring airflow pressure and vibration on the aircraft surface [5–7]. All these applications require proper design and functional safety of key components. Usually, several single sensors are employed, each of them dedicated to test only one parameter. However, this solution is costly, it occupies a large volume, and it may even affect the performance of original components [8]. Therefore, the development of a sensor that monitors dual parameters becomes reasonable. Several groups have started conducting research on dual-parameter sensors [9–12] in recent years. For example, the Precision Engineering Department of the Xi’jiao University has started its research on simultaneous pressure and temperature monitoring in 2006 [9], employing semiconductor Silicon technology. Nonetheless, silicon technology requires wire connection and can only operate up to 250°C, because of drastically increased leakage currents across the junction at elevated temperatures. The team led by Professor Albert P. Pisano started its research on blade temperature and pressure monitoring in gas engines in 2009 [10], based on the dual-LC resonant sensor. This technology showed great potential, because it could wirelessly measure multiple parameters without the need for a battery.The LC resonant sensor is a passive, wireless, easy-to-machine, adaptable to harsh environment [13, 14] sensor, based on a magnetic coupling readout method and combined with a ceramic micropackage technology. Dual LC circuits are usually integrated into a chip; however, changes in the coupling coefficient or in the LC circuit parameters influence the readout signal quality, which in turn affect the sensor-readout distance, as well as the sensor working temperature [15]. Therefore, it is important to analyze the coupling influence on the readout signal from a theoretical point of view, in order to build a foundation for the design and fabrication of an optimized dual-LC resonant sensor.We first introduce the single-parameter test concept. Then, we focus on the theory of dual LC circuits. Using the MATLAB software, the effect of changing either the coupling coefficient or the LC circuit parameters on the readout signal is analyzed. Finally, a test-system including a discrete capacitor soldered to a PCB based planar spiral coil is developed, and the sensor readout signal is analyzed under different relative positions among the coils. All experimental results are in agreement with the simulation results.


## 2. Single-Parameter Test

As shown in Figure1, to interrogate the sensor, a loop-antenna with inductance L0 is magnetically coupled to an inductance L1. The loop-antenna sends out an alternating current sweep frequency signal of a certain bandwidth. Using the transformer network theory and Kirchhoff law, the input impedance Zin looking into the antenna is given by [16](1)Zin=R0+jwL0+wM2jwL1+R1+1/jwC1,(2)M=kL0L1,(3)w=2πf,where k denotes the coupling coefficient between inductors and f represents frequency of AC signal. By analyzing (1), a simple equation for fmin, the minimum measured frequency extracted from the impedance-phase curve is obtained [17] as (4)fmin=f01+k24+18Q2,where Q denotes the LC circuit quality factor, while f0 is the self-resonant frequency and is defined below as(5)f0=12πL1C1.The equation for Zin is analyzed using the MATLAB software. Values for the initialized variables are listed in Table 1.Table 1
Initial values of the antenna and the LC circuit.
L
0 (µH)
R
0 (Ω)
L
1 (µH)
R
1 (Ω)
C
1 (pF)
k
1.5
3
6
3
10
0.1 : 0.05 : 0.3Figure 1
Schematic diagram of the single-parameter LC resonant sensor system.The coupling coefficientk in Table 1 is varied from 0.1 to 0.3, with a 0.05 step length. The other variables are kept constant. As shown in Figure 2, the minimum measured frequency fmin shifts rightward with a corresponding decreased phase dip Φmin, while the frequency band is clearly broadened. For k=0.2, the measured fmin is 20.74 MHz, Φmin is equal to −67.5°, while the self-resonant frequency f0 is calculated to be 20.54 MHz. The difference between fmin and f0 is found to be very small.Figure 2
Change of coupling coefficientk as a function of frequency.


## 3. Dual-Parameters Test

As shown in Figure3, the antenna is coupled with two LC circuits. The added LC circuit results in two additional couplings. Similar to the single-parameter test, the transformer network theory and Kirchhoff law are used, and the input impedance Zin looking into the antenna is given by [18](6)Zin=R0+jwL0+jwM1-w2M2M12-jwM1Z2Z1Z2+w2M122+jwM2-w2M1M12-jwM2Z1Z1Z2+w2M122,where (7)M1=k1L0L1,M2=k2L0L2,M12=k12L1L2.k1, k2, k12 are coupling coefficients between the antenna and inductor 1, antenna and inductor 2, and between inductors, respectively. Z1 and Z2 denote the system impedances of LC circuits 1 and 2 (Figure 3), respectively, which are defined below:(8)Z1=R1+jwL1+1jwC1,Z2=R2+jwL2+1jwC2.Figure 3
Test of an antenna and dual LC circuits.As seen from (6), the impedance becomes very complicated, and it is difficult to derive the relation between fmin1, fmin2 (measured frequencies) and f01, f02 (resonant frequencies) [18]. Using MATLAB, (6) is analyzed by initializing the variables, in order to see its influence on the final readout signal. Table 2 indicates that the variables listed in Table 1 remain even after the addition of LC circuit 2.Table 2
Initial values for the antenna and for the two LC circuits.
L
0 (µH)
R
0 (Ω)
L
1 (µH)
R
1 (Ω)
C
1 (pF)
L
2 (µH)
R
2 (Ω)
C
2 (pF)
k
1
k
2
k
12
1.5
3
6
3
10
6
3
10
0.2
0.2
0.2


### 3.1. Change in the LC Circuit Parameters

First, capacitanceC2 shown in Table 2 is changed from 3 pF to 9 pF, with a 2 pF step, as shown in Table 3.Table 3
C
2 change from 3 pF to 9 pF.
L
0 (µH)
R
0 (Ω)
L
1 (µH)
R
1 (Ω)
C
1 (pF)
L
2 (µH)
R
2 (Ω)
C
2 (pF)
k
1
k
2
k
12
1.5
3
6
3
10
6
3
3 : 2 : 9, 10 : 2 : 18
0.2
0.2
0.2The simulation results are shown in Figure4(a). While the measured frequency fmin2 (the right readout signal) shifts leftward, with a rapidly decreasing signal strength, the left readout signal only suffers from a slight change, simultaneously with the measured frequency fmin1 decreasing from 20.60 MHz to 19.49 MHz and the impedance phase Φmin1 dropping from −70.19° to −77.17°. It is found that the addition of LC circuit 2 to the coupling leads to a slight increase in the left readout signal strength, in contrast with the scenario without circuit 2 as shown in Figure 2.Figure 4
Simulation results upon changingC2 from (a) 3 pF to 9 pF, (b) 10 pF to 18 pF.
(a)
(b)Similarly, results in Figure4(b) show a situation where capacitance C2 is changed from 10 pF to 18 pF, with the other variables kept constant (Table 3). From Figure 4(b), we see that fmin2 (left readout signal) shifts leftward with a slightly increased signal strength, while the LC circuit 1 readout signal (right-hand side of the plot) clearly changes, with fmin1 decreasing from 22.18 MHz to 21.52 MHz and Φmin1 being enhanced from 45.83° to −32.00°. For C2=10pF, namely, when the parameters in the LC circuits are the same, the two readout signals merge into a single signal.From the results shown in Figure4, we state that in a dual-parameter test system, the change in the circuit parameter greatly affects its own readout signal, at the same time slightly influencing the other readout signal. Further, as indicated from Figure 4, when the coupling coefficients k1 and k2 are the same, and the more apart f01 is from f02, the higher is the averaged value of both signal strengths.


### 3.2. Change of Coupling for Circuits at the Same Frequency

It has already been shown that when the parameters in LC circuits are the same, namely,L1=L2, C1=C2, and k1=k2, there will be only one readout signal. Thus, in the following analysis, only the coupling coefficient k2 in Table 2 is changed from 0.05 to 0.25, with the other variables are shown in Table 4.Table 4
Change ofk2.
L
0 (µH)
R
0 (Ω)
L
1 (µH)
R
1 (Ω)
C
1 (pF)
L
2 (µH)
R
2 (Ω)
C
2 (pF)
k
1
k
2
k
12
1.5
3
6
3
10
6
3
10
0.2
0.05 : 0.05 : 0.25
0.2The simulation results presented in Figure5 show the situation when the coupling coefficients k1 and k2 are different, with the two readout signals clearly separated. With the increase in k2, the left signal varies similarly to what is described for the single-parameter condition, mentioned in Section 2. Whereas the right signal changes simultaneously, through analysis on the right signal, it is found that the readout signal strength is stronger when k2 is at a farther distance from k1.Figure 5
Simulation results upon changingk2.


### 3.3. Change of Coupling for Circuits at Different Frequencies

In contrast with data presented in Table2, in this section, the variable L2 is changed (Table 5). In this case, the self-resonant frequency fmin2 derived from (5) is 29.05 MHz. Similarly, only L2 in Table 5 is changed in order to track the influence on the readout signal in simulation.Table 5
Initial value of the antenna and two LC circuits.
L
0 (µH)
R
0 (Ω)
L
1 (µH)
R
1 (Ω)
C
1 (pF)
L
2 (µH)
R
2 (Ω)
C
2 (pF)
k
1
k
2
k
12
1.5
3
6
3
10
3
3
10
0.2
0.2
0.2First, variablek12 in Table 5 is changed from 0.1 to 0.5 with a 0.1 step, which is listed in Table 6.Table 6
Change ink12.
L
0 (µH)
R
0 (Ω)
L
1 (µH)
R
1 (Ω)
C
1 (pF)
L
2 (µH)
R
2 (Ω)
C
2 (pF)
k
1
k
2
k
12
1.5
3
6
3
10
3
3
10
0.2
0.2
0.1 : 0.1 : 0.5From Figure6, it is found that in this case, the readout signals move in an opposite direction to the increase in k12. The left frequency fmin1 changes from 20.67 MHz to 19.11 MHz, with a slightly decreased Φmin, while the right fmin2 increases from 29.55 MHz to 36.53 MHz, with a readout signal strength rapidly decreasing. Figure 6 shows that increasing k12 contributes to the separation of the two signals, and this leads to a strengthened and weakened readout signal.Figure 6
Simulation results upon changingk12.The coupling coefficientk1 in Table 5 is changed. As shown in Figure 7(a), with the decrease in k1, the left readout signal (LC circuit 1) is changed similar to the single-parameter situation, while the right signal fmin2 (LC circuit 2) shifts rightward 0.14 MHz with Φmin2 clearly strengthened from 9.09° to −45.01°.Figure 7
Simulation results upon changing (a)k1 and (b) k2.
(a)
(b)Finally, thek2 listed in Table 5 is also changed. As shown in Figure 7(b), with the increase in k2, the right readout signal (LC circuit 2) is changed, similarly to the single-parameter situation; however, the left signal slightly varies, with fmin1 increased by 0.07 MHz and Φmin1 enhanced by only 5°.From what was described in Figure7, with respect to changing k1 and k2, it is found that changing coupling between the LC circuit and antenna greatly affects its own readout signal, which is similar to the single-parameter situation (as stated in Section 2); however, it simultaneously influences the other readout signal. When f01 is smaller than f02, decreasing k1 will get an enhanced readout signal Φmin2, while increasing k2 will get a slightly strengthened Φmin1.


## 3.1. Change in the LC Circuit Parameters

First, capacitanceC2 shown in Table 2 is changed from 3 pF to 9 pF, with a 2 pF step, as shown in Table 3.Table 3
C
2 change from 3 pF to 9 pF.
L
0 (µH)
R
0 (Ω)
L
1 (µH)
R
1 (Ω)
C
1 (pF)
L
2 (µH)
R
2 (Ω)
C
2 (pF)
k
1
k
2
k
12
1.5
3
6
3
10
6
3
3 : 2 : 9, 10 : 2 : 18
0.2
0.2
0.2The simulation results are shown in Figure4(a). While the measured frequency fmin2 (the right readout signal) shifts leftward, with a rapidly decreasing signal strength, the left readout signal only suffers from a slight change, simultaneously with the measured frequency fmin1 decreasing from 20.60 MHz to 19.49 MHz and the impedance phase Φmin1 dropping from −70.19° to −77.17°. It is found that the addition of LC circuit 2 to the coupling leads to a slight increase in the left readout signal strength, in contrast with the scenario without circuit 2 as shown in Figure 2.Figure 4
Simulation results upon changingC2 from (a) 3 pF to 9 pF, (b) 10 pF to 18 pF.
(a)
(b)Similarly, results in Figure4(b) show a situation where capacitance C2 is changed from 10 pF to 18 pF, with the other variables kept constant (Table 3). From Figure 4(b), we see that fmin2 (left readout signal) shifts leftward with a slightly increased signal strength, while the LC circuit 1 readout signal (right-hand side of the plot) clearly changes, with fmin1 decreasing from 22.18 MHz to 21.52 MHz and Φmin1 being enhanced from 45.83° to −32.00°. For C2=10pF, namely, when the parameters in the LC circuits are the same, the two readout signals merge into a single signal.From the results shown in Figure4, we state that in a dual-parameter test system, the change in the circuit parameter greatly affects its own readout signal, at the same time slightly influencing the other readout signal. Further, as indicated from Figure 4, when the coupling coefficients k1 and k2 are the same, and the more apart f01 is from f02, the higher is the averaged value of both signal strengths.


## 3.2. Change of Coupling for Circuits at the Same Frequency

It has already been shown that when the parameters in LC circuits are the same, namely,L1=L2, C1=C2, and k1=k2, there will be only one readout signal. Thus, in the following analysis, only the coupling coefficient k2 in Table 2 is changed from 0.05 to 0.25, with the other variables are shown in Table 4.Table 4
Change ofk2.
L
0 (µH)
R
0 (Ω)
L
1 (µH)
R
1 (Ω)
C
1 (pF)
L
2 (µH)
R
2 (Ω)
C
2 (pF)
k
1
k
2
k
12
1.5
3
6
3
10
6
3
10
0.2
0.05 : 0.05 : 0.25
0.2The simulation results presented in Figure5 show the situation when the coupling coefficients k1 and k2 are different, with the two readout signals clearly separated. With the increase in k2, the left signal varies similarly to what is described for the single-parameter condition, mentioned in Section 2. Whereas the right signal changes simultaneously, through analysis on the right signal, it is found that the readout signal strength is stronger when k2 is at a farther distance from k1.Figure 5
Simulation results upon changingk2.


## 3.3. Change of Coupling for Circuits at Different Frequencies

In contrast with data presented in Table2, in this section, the variable L2 is changed (Table 5). In this case, the self-resonant frequency fmin2 derived from (5) is 29.05 MHz. Similarly, only L2 in Table 5 is changed in order to track the influence on the readout signal in simulation.Table 5
Initial value of the antenna and two LC circuits.
L
0 (µH)
R
0 (Ω)
L
1 (µH)
R
1 (Ω)
C
1 (pF)
L
2 (µH)
R
2 (Ω)
C
2 (pF)
k
1
k
2
k
12
1.5
3
6
3
10
3
3
10
0.2
0.2
0.2First, variablek12 in Table 5 is changed from 0.1 to 0.5 with a 0.1 step, which is listed in Table 6.Table 6
Change ink12.
L
0 (µH)
R
0 (Ω)
L
1 (µH)
R
1 (Ω)
C
1 (pF)
L
2 (µH)
R
2 (Ω)
C
2 (pF)
k
1
k
2
k
12
1.5
3
6
3
10
3
3
10
0.2
0.2
0.1 : 0.1 : 0.5From Figure6, it is found that in this case, the readout signals move in an opposite direction to the increase in k12. The left frequency fmin1 changes from 20.67 MHz to 19.11 MHz, with a slightly decreased Φmin, while the right fmin2 increases from 29.55 MHz to 36.53 MHz, with a readout signal strength rapidly decreasing. Figure 6 shows that increasing k12 contributes to the separation of the two signals, and this leads to a strengthened and weakened readout signal.Figure 6
Simulation results upon changingk12.The coupling coefficientk1 in Table 5 is changed. As shown in Figure 7(a), with the decrease in k1, the left readout signal (LC circuit 1) is changed similar to the single-parameter situation, while the right signal fmin2 (LC circuit 2) shifts rightward 0.14 MHz with Φmin2 clearly strengthened from 9.09° to −45.01°.Figure 7
Simulation results upon changing (a)k1 and (b) k2.
(a)
(b)Finally, thek2 listed in Table 5 is also changed. As shown in Figure 7(b), with the increase in k2, the right readout signal (LC circuit 2) is changed, similarly to the single-parameter situation; however, the left signal slightly varies, with fmin1 increased by 0.07 MHz and Φmin1 enhanced by only 5°.From what was described in Figure7, with respect to changing k1 and k2, it is found that changing coupling between the LC circuit and antenna greatly affects its own readout signal, which is similar to the single-parameter situation (as stated in Section 2); however, it simultaneously influences the other readout signal. When f01 is smaller than f02, decreasing k1 will get an enhanced readout signal Φmin2, while increasing k2 will get a slightly strengthened Φmin1.


## 4. Experiment and Test

The coupling coefficient has a connection with many other factors such as a spiral coil shape, position, gap, slant angle, facing area between coils, and so forth. There are many studies describing the method to exactly calculate the coupling coefficient between two spiral coils in theory [19–21]; however, in our analysis, this is not required because of the complicated mathematical expressions involved. It is well known that the coupling coefficient is monotonic with the facing area and the gap between two planar spiral coils [22]; hence, for qualitative analysis, all coils in the experiment are printed on a PCB board with the same dimensions (Table 7). The LC circuit comprises a capacitor with pins soldered to the PCB board.Table 7
Dimensions of the PCB-based planar spiral coil.
Outer dimension of inductor (mm2)
Number of turns
Wire width (mm)
Wire spacing (mm)
Wire thickness (µm)
Substrate thickness (mm)
Substrate dimension (mm2)
3
0
×
30
10
0.3
0.3
15
0.58
33
×
33


### 4.1. Relative Position 1

As visible from Figure8, the antenna is placed in the middle and the LC circuits are positioned to the left and to the right sides of it, in perfect alignment with the antenna. Both the distances d1 and d2 are equal to 7 mm, to ensure k1=k2. Capacitance C1, soldered to the LC circuit 1 is changed from 3 pF to 9 pF and from 10 pF to 18 pF, with a step length of 2 pF, respectively, while C2 stays fixed at 10 pF. The test result is shown in Figure 9. It is found that a strong readout signal and a weak readout signal are clearly distinguishable when C1 deviates from C2, whereas they merge into one signal when C1=C2. Thus, it is better to deviate C1 from C2 to ensure that both signals are strengthened. The average strength of the signal when C2 deviates from C1 agrees with that obtained from the MATLAB simulation results.Figure 8
Relative position of three coils.Figure 9
Test result upon changingC1 from (a) 3 pF to 9 pF, (b) 10 pF to 18 pF.
(a)
(b)Next, we move coil 1 leftward from 4 mm to 10 mm with a 1 mm-step, whiled2 is kept fixed at 7 mm. The capacitors soldered are both 10 pF. It is known that moving inductor 1 causes a change in both k1 and k12, but the change in k1 is larger than the one in k12. In fact, for a magnetic field, the strength is in inverse ratio with the third power of the gap [22], while d12 is almost twice the distance of d1. The result is shown in Figure 10. It is found that the right signal strength gradually decreases from 4 mm to 7 mm, and it completely disappears at 7 mm. For longer distances, the right signal starts to increase again from 7 mm to 10 mm. This proves that the signal weakens when k1 gets close to k2, while it gets stronger when they deviate from one another. The left test signal changes similar to the single-parameter test situation.Figure 10
Test result upon moving coil 1 leftward.To study the influence of different resonant frequency LC circuits on the antenna, a capacitor of 20 pF is soldered to circuit 1, whileC2 is kept at 10 pF. Coil 1 is moved leftward from 3 mm to 6 mm, while d2 is kept at 7 mm. The readout result is shown in Figure 11(a). It is found that when f01 is smaller than f02, the right strength (LC circuit 2) is clearly enhanced, and the left signal (LC circuit 1) varies similar to the single-parameter situation. Then, as above, coil 2 is moved leftward from 6 mm to 3 mm, while d1 is kept at 7 mm. The test result shown in Figure 11(b) indicates that the left signal is slightly enhanced differently from the situation described in Figure 11(a).Figure 11
Test result of moving (a) coil 1 outward, (b) coil 2 inward.
(a)
(b)


### 4.2. Relative Position 2

In the following configuration, the antenna is placed on the left side and three coils are in perfect mutual alignment. The capacitors soldered to the circuits are both 10 pF. Coil 2 in the experiment is moved outward from 0.5 mm to 3 mm, with a 0.5 mm step, whiled1 is kept at 7 mm. As above, both couplings k2 and k12 are decreased when moving coil 2 rightward; however, k12 increases considerably more than k2, for the gap d12 is much smaller than d2. The result is shown in Figure 12(b). It is found that the two test signals move close to each other, which proves that decreasing k12 contributes to the two signals approach.Figure 12
(a) Relative position of three coils. (b) Test result.
(a)
(b)Then, as shown in Figure13(a), coil 2 is moving in the Y-direction with a step of 7.5 mm, to gradually stagger two coils. The gaps d12 and d1 are kept equal to 1 mm and 7 mm, respectively. The results are shown in Figure 13(b). It is found that at the beginning the two test signals move close to each other up to 22.5 mm, because of the decrease in the k12 coupling. Next, the coupling between LC circuits becomes very weak, causing tested signals to slightly change until coil 2 gets totally away from coil 1.Figure 13
(a) Relative position of the three coils. (b) Test result.
(a)
(b)


## 4.1. Relative Position 1

As visible from Figure8, the antenna is placed in the middle and the LC circuits are positioned to the left and to the right sides of it, in perfect alignment with the antenna. Both the distances d1 and d2 are equal to 7 mm, to ensure k1=k2. Capacitance C1, soldered to the LC circuit 1 is changed from 3 pF to 9 pF and from 10 pF to 18 pF, with a step length of 2 pF, respectively, while C2 stays fixed at 10 pF. The test result is shown in Figure 9. It is found that a strong readout signal and a weak readout signal are clearly distinguishable when C1 deviates from C2, whereas they merge into one signal when C1=C2. Thus, it is better to deviate C1 from C2 to ensure that both signals are strengthened. The average strength of the signal when C2 deviates from C1 agrees with that obtained from the MATLAB simulation results.Figure 8
Relative position of three coils.Figure 9
Test result upon changingC1 from (a) 3 pF to 9 pF, (b) 10 pF to 18 pF.
(a)
(b)Next, we move coil 1 leftward from 4 mm to 10 mm with a 1 mm-step, whiled2 is kept fixed at 7 mm. The capacitors soldered are both 10 pF. It is known that moving inductor 1 causes a change in both k1 and k12, but the change in k1 is larger than the one in k12. In fact, for a magnetic field, the strength is in inverse ratio with the third power of the gap [22], while d12 is almost twice the distance of d1. The result is shown in Figure 10. It is found that the right signal strength gradually decreases from 4 mm to 7 mm, and it completely disappears at 7 mm. For longer distances, the right signal starts to increase again from 7 mm to 10 mm. This proves that the signal weakens when k1 gets close to k2, while it gets stronger when they deviate from one another. The left test signal changes similar to the single-parameter test situation.Figure 10
Test result upon moving coil 1 leftward.To study the influence of different resonant frequency LC circuits on the antenna, a capacitor of 20 pF is soldered to circuit 1, whileC2 is kept at 10 pF. Coil 1 is moved leftward from 3 mm to 6 mm, while d2 is kept at 7 mm. The readout result is shown in Figure 11(a). It is found that when f01 is smaller than f02, the right strength (LC circuit 2) is clearly enhanced, and the left signal (LC circuit 1) varies similar to the single-parameter situation. Then, as above, coil 2 is moved leftward from 6 mm to 3 mm, while d1 is kept at 7 mm. The test result shown in Figure 11(b) indicates that the left signal is slightly enhanced differently from the situation described in Figure 11(a).Figure 11
Test result of moving (a) coil 1 outward, (b) coil 2 inward.
(a)
(b)


## 4.2. Relative Position 2

In the following configuration, the antenna is placed on the left side and three coils are in perfect mutual alignment. The capacitors soldered to the circuits are both 10 pF. Coil 2 in the experiment is moved outward from 0.5 mm to 3 mm, with a 0.5 mm step, whiled1 is kept at 7 mm. As above, both couplings k2 and k12 are decreased when moving coil 2 rightward; however, k12 increases considerably more than k2, for the gap d12 is much smaller than d2. The result is shown in Figure 12(b). It is found that the two test signals move close to each other, which proves that decreasing k12 contributes to the two signals approach.Figure 12
(a) Relative position of three coils. (b) Test result.
(a)
(b)Then, as shown in Figure13(a), coil 2 is moving in the Y-direction with a step of 7.5 mm, to gradually stagger two coils. The gaps d12 and d1 are kept equal to 1 mm and 7 mm, respectively. The results are shown in Figure 13(b). It is found that at the beginning the two test signals move close to each other up to 22.5 mm, because of the decrease in the k12 coupling. Next, the coupling between LC circuits becomes very weak, causing tested signals to slightly change until coil 2 gets totally away from coil 1.Figure 13
(a) Relative position of the three coils. (b) Test result.
(a)
(b)


## 5. Conclusion

This paper first introduced the theoretical model of a dual-parameter LC resonant sensor. Because of the difficulties in clearly analyzing complicate mathematical relations, a MATLAB software is used. We initialized the variables and studied the influence of changing variables on the readout signal according to different situations. Preliminarily conclusions are drawn based on the above-mentioned MATLAB tool. Finally, a test system including a discrete capacitor soldered to a PCB-based planar spiral coil is built and different relative positions among three spiral coils were researched. All experimental results are in agreement with simulation results in MATLAB.



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*Source: 437869-2015-05-25.xml*