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

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 f 01 = f 02 , it is better to broaden the difference between the two coupling coefficients k 1 and k 2 . On the other side, whenf 01 is smaller thanf 02 , it is better to decrease the coupling between sensor inductors k 12 , 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 agreementwith the results of theMATLAB simulation.


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][2][3], temperature and vibration monitoring in key components of coal mining machines [4], and monitoring airflow pressure and vibration on the aircraft surface [5][6][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 dualparameter sensors [9][10][11][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-tomachine, adaptable to harsh environment [13,14] sensor, 2 Advances in Mathematical Physics 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 sensorreadout 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.

Single-Parameter Test
As shown in Figure 1, to interrogate the sensor, a loopantenna with inductance  0 is magnetically coupled to an inductance  1 .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  in 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  min , the minimum measured frequency extracted from the impedance-phase curve is obtained [17] as where  denotes the LC circuit quality factor, while  0 is the self-resonant frequency and is defined below as The equation for  in is analyzed using the MATLAB software.
Values for the initialized variables are listed in Table 1.
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  min shifts rightward with a corresponding decreased phase dip Φ min , while the frequency band is clearly broadened.For  = 0.2, the measured  min is 20.74 MHz,  Φ min is equal to −67.5 ∘ , while the self-resonant frequency  0 is calculated to be 20.54MHz.The difference between  min and  0 is found to be very small.

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  in looking into the antenna is given by [18] where 1 ,  2 ,  12 are coupling coefficients between the antenna and inductor 1, antenna and inductor 2, and between inductors,  respectively. 1 and  2 denote the system impedances of LC circuits 1 and 2 (Figure 3), respectively, which are defined below: As seen from ( 6), the impedance becomes very complicated, and it is difficult to derive the relation between  min1 ,  min2 (measured frequencies) and  01 ,  02 (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. 2 is changed from 3 pF to 9 pF, with a 2 pF step, as shown in Table 3.

Change in the LC
The simulation results are shown in Figure 4(a).While the measured frequency  min2 (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  min1 decreasing from 20.60 MHz to 19.49 MHz and the impedance phase Φ min1 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.
Similarly, results in Figure 4(b) show a situation where capacitance  2 is changed from 10 pF to 18 pF, with the other variables kept constant (Table 3).From Figure 4(b), we see that  min2 (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  min1 decreasing from 22.18 MHz to 21.52 MHz and Φ min1 being enhanced from 45.83 ∘ to −32.00 ∘ .For  2 = 10 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 dualparameter 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  1 and  2 are the same, and the more apart  01 is from  02 , the higher is the averaged value of both signal strengths.

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,  1 =  2 ,  1 =  2 , and  1 =  2 , there will be only one readout signal.Thus, in the following analysis, only the coupling coefficient  2 in Table 2 is changed from 0.05 to 0.25, with the other variables are shown in Table 4.
The simulation results presented in Figure 5 show the situation when the coupling coefficients  1 and  2 are different, with the two readout signals clearly separated.With the increase in  2 , 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  2 is at a farther distance from  1 .

Change of Coupling for Circuits at Different Frequencies.
In contrast with data presented in Table 2, in this section, the variable  2 is changed (Table 5).In this case, the self-resonant frequency  min2 derived from (5) is 29.05 MHz.Similarly, only  2 in Table 5 is changed in order to track the influence on the readout signal in simulation.
First, variable  12 in Table 5 is changed from 0.1 to 0.5 with a 0.1 step, which is listed in Table 6.
From Figure 6, it is found that in this case, the readout signals move in an opposite direction to the increase in  12 .The left frequency  min1 changes from 20.67 MHz to    19.11 MHz, with a slightly decreased Φ min , while the right  min2 increases from 29.55 MHz to 36.53 MHz, with a readout signal strength rapidly decreasing.Figure 6 shows that increasing  12 contributes to the separation of the two signals, and this leads to a strengthened and weakened readout signal.
The coupling coefficient  1 in Table 5 is changed.As shown in Figure 7(a), with the decrease in  1 , the left readout signal (LC circuit 1) is changed similar to the singleparameter situation, while the right signal  min2 (LC circuit 2) shifts rightward 0.14 MHz with Φ min2 clearly strengthened from 9.09 ∘ to −45.01 ∘ .
Finally, the  2 listed in Table 5 is also changed.As shown in Figure 7(b), with the increase in  2 , the right readout signal (LC circuit 2) is changed, similarly to the single-parameter situation; however, the left signal slightly varies, with  min1 increased by 0.07 MHz and Φ min1 enhanced by only 5 ∘ .From what was described in Figure 7, with respect to changing  1 and  2 , 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  01 is smaller than  02 , decreasing  1 will get an enhanced readout signal Φ min2 , while increasing  2 will get a slightly strengthened Φ min1 .

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][20][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.

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  1 and  2 are equal to 7 mm, to ensure  1 =  2 .Capacitance  1 , 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  2 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  1 deviates from  2 , whereas they merge into one signal when  1 =  2 .Thus, it is better to deviate  1 from  2 to ensure that both signals are strengthened.The average strength of the signal when  2 deviates from  1 agrees with that obtained from the MATLAB simulation results.
Next, we move coil 1 leftward from 4 mm to 10 mm with a 1 mm-step, while  2 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  1 and  12 , but the change in  1 is larger than the one in  12 .In fact, for a magnetic field, the strength is in inverse ratio with the third power of the gap [22], while  12 is almost twice the distance of  1 .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  1 gets close to  2 , while it gets stronger when they deviate from one another.The left test signal changes similar to the single-parameter test situation.
To study the influence of different resonant frequency LC circuits on the antenna, a capacitor of 20 pF is soldered to circuit 1, while  2 is kept at 10 pF.Coil 1 is moved leftward from 3 mm to 6 mm, while  2 is kept at 7 mm.The readout result is shown in Figure 11(a).It is found that when  01 is smaller than  02 , 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  1 is kept at 7 mm.The test result shown in Figure 11 is slightly enhanced differently from the situation described in Figure 11(a).

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.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  12 and  1 are kept equal to 1 mm and 7 mm, respectively.The results are shown in Figure 13 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.

Conclusion
This paper first introduced the theoretical model of a dualparameter LC resonant sensor.Because of the difficulties in clearly analyzing complicate mathematical relations, a MAT-LAB 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.

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

Figure 3 :
Figure 3: Test of an antenna and dual LC circuits.

Table 2 :
Initial values for the antenna and for the two LC circuits.

Table 5 :
Initial value of the antenna and two LC circuits.

Table 7 :
Dimensions of the PCB-based planar spiral coil.
5 mm step, while  1 is kept at 7 mm.As above, both couplings  2 and  12 are decreased when moving coil 2 rightward; however,  12 increases considerably more than  2 , for the gap  12 is much smaller than  2 .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  12 contributes to the two signals approach.