Advances in Mathematical Physics
Volume 2015 (2015), Article ID 437869, 9 pages
http://dx.doi.org/10.1155/2015/437869

Research Article

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

Jijun Xiong,1,2 Tanyong Wei,1,2 Tao Luo,1,2 Qiulin Tan,1,2,3 Chenyang Xue,1,2 Jun Liu,1,2 and Wendong Zhang1,2

¹Key Laboratory of Instrumentation Science & Dynamic Measurement, Ministry of Education, North University of China, Tai Yuan 030051, China
²Science and Technology on Electronic Test & Measurement Laboratory, North University of China, Tai Yuan 030051, China
³State Key Laboratory of Transducer Technology, Department of Precision Instrument and Mechanology, Tsinghua University, Beijing 10084, China

Received 1 February 2015; Accepted 3 May 2015

Academic Editor: Mikhail Panfilov

Copyright © 2015 Jijun Xiong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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. When , it is better to broaden the difference between the two coupling coefficients and . On the other side, when is smaller than , it is better to decrease the coupling between sensor inductors , 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.

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 Figure 1, to interrogate the sensor, a loop-antenna with inductance is magnetically coupled to an inductance . 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 looking into the antenna is given by [16]

where denotes the coupling coefficient between inductors and represents frequency of AC signal. By analyzing (1), a simple equation for , the minimum measured frequency extracted from the impedance-phase curve is obtained [17] as

where denotes the LC circuit quality factor, while is the self-resonant frequency and is defined below as

The equation for 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.

(H)
(Ω)
(H)
()
(pF)

1.5
3
6
3
10
0.1 : 0.05 : 0.3

Figure 1: Schematic diagram of the single-parameter LC resonant sensor system.

The coupling coefficient 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 shifts rightward with a corresponding decreased phase dip , while the frequency band is clearly broadened. For , the measured is 20.74 MHz, is equal to −67.5°, while the self-resonant frequency is calculated to be 20.54 MHz. The difference between and is found to be very small.

Figure 2: Change of coupling coefficient as a function of frequency.

3. Dual-Parameters Test

As shown in Figure 3, 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 looking into the antenna is given by [18]

where

, , are coupling coefficients between the antenna and inductor 1, antenna and inductor 2, and between inductors, respectively. and denote the system impedances of LC circuits 1 and 2 (Figure 3), respectively, which are defined below:

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 , (measured frequencies) and , (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.

(H)
(Ω)
(µH)
(Ω)
(pF)
(µH)
(Ω)
(pF)

1.5
3
6
3
10
6
3
10
0.2
0.2
0.2

3.1. Change in the LC Circuit Parameters

First, capacitance shown in Table 2 is changed from 3 pF to 9 pF, with a 2 pF step, as shown in Table 3.

Table 3: change from 3 pF to 9 pF.

(µH)
(Ω)
(µH)
(Ω)
(pF)
(µH)
(Ω)
(pF)

1.5
3
6
3
10
6
3
3 : 2 : 9, 10 : 2 : 18
0.2
0.2
0.2

The simulation results are shown in Figure 4(a). While the measured frequency (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 decreasing from 20.60 MHz to 19.49 MHz and the impedance phase 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.

(a)

(b)

Figure 4: Simulation results upon changing from (a) 3 pF to 9 pF, (b) 10 pF to 18 pF.

Similarly, results in Figure 4(b) show a situation where capacitance is changed from 10 pF to 18 pF, with the other variables kept constant (Table 3). From Figure 4(b), we see that (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 decreasing from 22.18 MHz to 21.52 MHz and being enhanced from 45.83° to −32.00°. For pF, 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 Figure 4, 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 and are the same, and the more apart is from , 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, , , and , there will be only one readout signal. Thus, in the following analysis, only the coupling coefficient in Table 2 is changed from 0.05 to 0.25, with the other variables are shown in Table 4.

Table 4: Change of .

(µH)
(Ω)
(µH)
(Ω)
(pF)
(µH)
(Ω)
(pF)

1.5
3
6
3
10
6
3
10
0.2
0.05 : 0.05 : 0.25
0.2

The simulation results presented in Figure 5 show the situation when the coupling coefficients and are different, with the two readout signals clearly separated. With the increase in , 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 is at a farther distance from .

Figure 5: Simulation results upon changing .

3.3. Change of Coupling for Circuits at Different Frequencies

In contrast with data presented in Table 2, in this section, the variable is changed (Table 5). In this case, the self-resonant frequency derived from (5) is 29.05 MHz. Similarly, only 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.

(µH)
(Ω)
(µH)
(Ω)
(pF)
(µH)
(Ω)
(pF)

1.5
3
6
3
10
3
3
10
0.2
0.2
0.2

First, variable 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 in .

(µH)
(Ω)
(µH)
(Ω)
(pF)
(µH)
(Ω)
(pF)

1.5
3
6
3
10
3
3
10
0.2
0.2
0.1 : 0.1 : 0.5

From Figure 6, it is found that in this case, the readout signals move in an opposite direction to the increase in . The left frequency changes from 20.67 MHz to 19.11 MHz, with a slightly decreased , while the right increases from 29.55 MHz to 36.53 MHz, with a readout signal strength rapidly decreasing. Figure 6 shows that increasing contributes to the separation of the two signals, and this leads to a strengthened and weakened readout signal.

Figure 6: Simulation results upon changing .

The coupling coefficient in Table 5 is changed. As shown in Figure 7(a), with the decrease in , the left readout signal (LC circuit 1) is changed similar to the single-parameter situation, while the right signal (LC circuit 2) shifts rightward 0.14 MHz with clearly strengthened from 9.09° to −45.01°.

(a)

(b)

Figure 7: Simulation results upon changing (a) and (b) .

Finally, the listed in Table 5 is also changed. As shown in Figure 7(b), with the increase in , the right readout signal (LC circuit 2) is changed, similarly to the single-parameter situation; however, the left signal slightly varies, with increased by 0.07 MHz and enhanced by only 5°.

From what was described in Figure 7, with respect to changing and , 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 is smaller than , decreasing will get an enhanced readout signal , while increasing will get a slightly strengthened .

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)

10
0.3
0.3
15
0.58

4.1. Relative Position 1

As visible from Figure 8, 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 and are equal to 7 mm, to ensure . Capacitance , 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 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 deviates from , whereas they merge into one signal when . Thus, it is better to deviate from to ensure that both signals are strengthened. The average strength of the signal when deviates from agrees with that obtained from the MATLAB simulation results.

Figure 8: Relative position of three coils.

(a)

(b)

Figure 9: Test result upon changing from (a) 3 pF to 9 pF, (b) 10 pF to 18 pF.

Next, we move coil 1 leftward from 4 mm to 10 mm with a 1 mm-step, while 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 and , but the change in is larger than the one in . In fact, for a magnetic field, the strength is in inverse ratio with the third power of the gap [22], while is almost twice the distance of . 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 gets close to , 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, while is kept at 10 pF. Coil 1 is moved leftward from 3 mm to 6 mm, while is kept at 7 mm. The readout result is shown in Figure 11(a). It is found that when is smaller than , 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 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).

(a)

(b)

Figure 11: Test result of moving (a) coil 1 outward, (b) coil 2 inward.

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, while is kept at 7 mm. As above, both couplings and are decreased when moving coil 2 rightward; however, increases considerably more than , for the gap is much smaller than . 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 contributes to the two signals approach.

(a)

(b)

Figure 12: (a) Relative position of three coils. (b) Test result.

Then, as shown in Figure 13(a), coil 2 is moving in the -direction with a step of 7.5 mm, to gradually stagger two coils. The gaps and 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 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.

(a)

(b)

Figure 13: (a) Relative position of the three coils. (b) Test result.

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.

Conflict of Interests

The authors declare no conflict of interests.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 61471324), the National Outstanding Youth Fund of China (no. 51425505), the Program for the Top Young Academic Leaders of Higher Learning Institutions of Shanxi Province, China, and the Post-doctor special Foundation of Tsinghua University, China (2014T70074).

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