Identification of Q value and angular frequency of smart gear antenna using the resonant return loss of the receiving antenna

Abstract

A wireless smart gear system has been being developed as a gear health monitoring system. This system works based on the magnetic coupling between its two major elements consisting of a monitoring antenna and a smart gear. In operation, the smart gear works as a working gear in the power transmission system while the monitoring antenna monitors this gear continuously. Specifically, since the monitoring antenna is fixed and kept connecting to a network analyzer during the gear operation, it allows the magnetic coupling to occur and remain thoroughly. As a result, the return loss signal of the monitoring antenna becomes a resonant return loss data of the smart gear system. In this paper, the authors propose a method to use this resonant return loss signal for identifying two parameters called the $Q$ value and the angular frequency of the smart gear side antenna. There are three cases of magnetic coupling that have been considered. These cases are called Open, Short, and Loaded by a 50 [Ω] resistor where the characteristics of the antenna on the smart gear side are changed via its connector by leaving Open or connecting to a Short and a 50 [Ω] resistor modules, respectively. The results indicate that the method is effective for parameter identification of the $Q$ value and the angular frequency of the smart gear side antenna for all three cases. Moreover, the analysis of the validated $Q$ value and the angular frequency also indicates the relation between these two characteristic values and the practical conditions of the smart gear. This finding consolidates the effectiveness of our developing wireless smart gear system.

Article Highlights

• This article proposes a parameter identification method for a developing smart gear health monitoring system. By only using the resonant return loss data obtained from the monitoring antenna, there are two characteristic values of the smart gear have been achieved.

• This article also introduces a completed return loss equation of a magnetically coupled system. This return loss equation is helpful for the computation of characteristic parameters of the magnetically coupled system such as a wireless power transfer system or a coupled antenna system.

• The results figure out that the identified characteristic parameters and the physical condition of the smart gear side’s antenna circuit are related to each other. This finding affirms the effectiveness of not only the proposed parameter identification method but also the developing smart gear system.

1 Introduction

Today, condition monitoring (CM) for gears in power transmission compels more and more attraction from researchers and scientists in the mechanical engineering field. A literature review shows that a number of interesting CM methods for gears have been proposed. Studies summarized by [1,2,3] have briefly revealed that scrutinizing and inspecting vibration data collected for the targeted gears are typically popular approaches for gear health condition monitoring in rotating machinery. However, the CM techniques that are based on the analysis of vibration data are never considered simple techniques. This comes mainly from the quality of the vibration data. Specifically, to extract the acoustic response or vibration data, transducers or accelerometers or even sensors are typically used. Due to the limitation of technology, these devices are normally attached to the surfaces of the housing or bearings holder instead of the targeted operation gears. In this way, this signal data contains the characteristic vibration data of not only the targeted gears but also the other elements such as the shafts or bearings of the mechanical system. Certainly, these described characteristic vibration data of the unrelated elements become a burden for the step of signal data processing. In other words, the signal data processing for gear characteristics challenges both the computational system and researchers due to the requirement for the removal of these noises. Regarding noise removal, [1,2,3] also indicated that noise removal is a sophisticated process. Typically, a signal averaging process called the time-synchronous averaging (TSA) technique has been known as the most popular technique to remove the background noise, especially for the data extracted from rotating machinery or a gearbox. In fact, the essential steps and techniques that must be implemented to achieve the results of the vibration–based gear health monitoring method are included in two typical different groups. These are the time-statistical analysis and the time–frequency analysis. Overall, the mentioned techniques and methods can be considered complicated for any system operator who does not have strong background knowledge and experience in the signal processing field. Besides, the application of this signal processing technique for the requirement of real-time detection is also a big challenge for the system designer and operator.

A recent literature review has given that, there are numerous interesting researches related to the topics of gear health monitoring have been introduced. Indeed, to enhance the vibration-based gear health monitoring technique, the proposed methods by [4, 5] showed a combination of the vibration data analysis and a convolutional neuron network technique for the recognition of cracks on the plastic gear of a gear meshing. The research by [6] tried to combine acoustic emission and vibration signal processing for monitoring gears in a power gearbox. With an aim to improve the gear macro-pitting detection in a gearbox [7] focused on the efficiency of a mixture of two methods that are vibration data processing and online visual monitoring of particles in lubricants. Currently, the use of sensors or sensitive materials for gear health monitoring purposes is also widespread. The study by [8] attempted to use eddy current sensors for the diagnosis of the failure of the gear teeth. [9] installed wireless temperature nodes on the housing of a gearbox for monitoring common potential faults such as gear teeth damage or oil level. Regarding the use of the micro-electro-mechanical system (MEMS), [10] integrated the micro-electro-mechanical system (MEMS) sensors on a pre-notched OH-58C spiral-bevel pinion gear of a helicopter for inspection of gear tooth faults. And [11] conducted an experiment to monitor the physical condition of a gear where the MEMS sensors were mounted directly on this gear.

Based on the literature work above, it is simple to realize that the gear health monitoring methods that are based on the vibrational signal processing technique are popular. The results of the research by [4,5,6,7], especially by [4, 5], show that the proposed methods are effective for the purposes of determining gear failures. Moreover, the authors of these studies have also indicated that compared to the classic plain vibration-based gear health monitoring techniques, the accuracy of their condition monitoring results was enhanced significantly. Nevertheless, since the vibration signal processing technique is also a sophisticated technique with numerous signal processing steps, the combination of this technique and the other convolutional neuron network technique or the acoustic emission basically increases the number of steps for reliable results. In the case of the techniques that use sensors and smart materials, the results of [8] confirmed that the effectiveness of the eddy current sensors is undeniable. The attached sensors on the idler gear were able to detect pitting on the face of the metal gear but the accuracy of the result of this study [8] depends considerably on the metallic material of the gear and the density of sensors on the idler gear. In comparison to [8], although the application of wireless sensor nodes in [9] can reduce the general dimensions of the gear health monitoring system remarkably, the complexity of the power supply module requires meticulous consideration before the application. Probably, the use of MEMS sensors can be considered a high-accuracy method. In the case of [10], the MEMS sensors were able to send the indication of gear failure until the end of the test. Besides, the spectrum data collected by these MEMS sensors was also helpful for the gear failure detection process. The results of [11] were the vibration response collected directly by the MEMS sensors. The advantage of this vibration data is that it has higher quality than the vibration data obtained via a piezo accelerometer or sensors mounted on the housing of the gearbox. Specifically, since the MEMS sensor can be integrated directly into the targeted gears, its vibration will have low noise from other machinery parts compared to the vibration data gained via the piezo accelerometer. Even if a condition monitoring method by attaching a microcontroller with embedded MEMS to the gears could be realized in an operational test in a laboratory environment, there are still several issues to be addressed. For example, the mass balance of gears rotating at high speeds is disturbed. Increasing the sampling frequency to measure meshing vibration requires an expensive and large microcontroller, which increases cost and installation space, and also increases power consumption. Therefore, a system directly mounted on gears should be simpler, lower cost, and smaller in volume.

To address these requirements, the authors have developed a new gear health monitoring system called a “smart gear” system. This system has two main components that are a smart gear and a monitoring antenna. Here, the smart gear is basically the operation gear that has a conductive sensor antenna pattern on its side surface. This conductive sensor pattern forms a sensor track circuit. The sintered conductive sensor locates at the tooth root area of all the gear teeth. Since the tooth root area of the gear is the place where the crack typically appears, the physical condition of the smart gear during the gear operation can affect the conductivity of the sensor directly. In other words, the physical condition of the sensor track circuit is also changed accordingly. The monitoring antenna has a conductive antenna pattern on its surface. Similarly, this conductive antenna pattern forms an antenna circuit. In the smart gear system, the monitoring antenna must monitor the physical condition of the smart gear via magnetic coupling between the sensor track circuit and the antenna circuit. During the gear operation, the monitoring antenna is connected to a network analyzer and placed nearby the smart gear to enable the magnetic coupling to occur.

In comparison with the introduced gear health monitoring methods above, the authors’ smart gear system has a simple structure and operation. The conductive sensor layer on the smart gear is created based on our self-developed laser sintering technique. This layer is just a thin layer of sintered silver nanoparticle ink. It means that there is no requirement for periodical maintenance or repair. This layer is designed as it will be broken when the smart gear is broken due to the gear operation. Thanks to the magnetic coupling, the physical condition of the smart gear can be observed via a computer that is connected to the network analyzer. Specifically, the return loss of the monitoring antenna will become a resonant return loss data when the magnetic coupling occurs. Our empirical work has proven that the change in the physical condition of the sensor track circuit results in the corresponding change of the magnetic coupling that can be observed via the resonant return loss of the monitoring antenna. Specifically, [12] specified that the conductive states of the sensor such as uncracked—healthy or cracked—unhealthy are related to the change of the resonant return loss shape of the monitoring antenna.

Nonetheless, although the founding of [12] revealed that the observation of the return loss shapes of the monitoring antenna can indicate the physical conditions of the smart gear, the characteristic values of the conductive sensor track circuit of the smart gear remain unknown. With a motivation to identify the characteristic information of the sensor track circuit of the smart gear wirelessly, this research proposes a method called the “characteristic parameter identification method.” The input data source of this method is the resonant return loss of the monitoring antenna. A return loss equation system has been established for the computation. In the scope of this research, there are only two parameters that are the Q factor and the natural angular frequency of the smart gear side sensor track circuit, here and after called “$Q_{2}$” and “$\omega_{2}$”, respectively, are identified. This comes from the fact that the number of unknown variables including “$Q_{2}$” and “$\omega_{2}$” in the established return loss equation system exceeds the number of equations. The result demonstrates that the proposed method is effective for parameter identification of the sensor track circuit of the smart gear merely based on the resonant return loss of the monitoring antenna.

The structure of this paper consists of 4 sections. The introduction section describes the background, motivation, and objective of the research. In the second section, the authors provide the materials and methods used in this research. A flow chart of research work is also provided in this section. Next, the third section explains and discusses the results of identified “$Q_{2}$” and “$\omega_{2}$”. Finally, section fourth provides the conclusions and future works of the research.

2 Models and methodology

This section includes the description of components in our developing smart gear system, the experimental samples, and the proposed methodology of this research.

2.1 The smart gear system and models

Figure 1 illustrates two practical components of our developing smart gear system and its assembling into a driving test rig. The two components including a monitoring antenna and a smart gear are denoted in Fig. 1a. In this figure, the monitoring antenna is on the left side and the smart gear is on the right side, respectively. As can be seen, the patterns of the monitoring antenna and the smart gear share a similar antenna design. This antenna design is sketched in reference to a spur gear with a face-width being 10 [mm], a module being 1 [mm], and a number of gear teeth being 48. The antenna design comprises two main parts: an inner spiral coil and an outer annulus ground (GND) band. The inner spiral coil is based on Archimedes’ spiral interpolation that has two turns with the pitch, initial radius, and line width are 1 [mm], 12.5 [mm], and 0.5 [mm], respectively. This coil plays a role as a spiral antenna for magnetic coupling. The dimensions of the outer annulus ground (GND) band are 31 [mm] for the inner diameter and 42 [mm] for the outer diameter, correspondingly. Figure 1b shows how a practical smart gear health monitoring system works. As is shown, the monitoring antenna is held in front of the smart gear side surface where the sensor layer is located. The smart gear is engaged to a master gear playing a role as an operation gear. Thanks to a holder made of ABS plastic, the monitoring antenna is kept fixed and parallel to the sensor layer of the smart gear. Since the monitoring antenna is connected to a network analyzer via a radio frequency (RF) cable during the gear operation, this setting allows the magnetic coupling to occur between the antenna circuit of the monitoring antenna and the sensor track circuit of the smart gear continuously even if the smart gear is running or stopped. In this way, the operator of the smart gear system can monitor any change in the characteristics of the sensor track circuit via the resonant return loss of the monitoring antenna as discussed earlier in the introduction section.

Fig. 1

Smart gear health monitoring system: a monitoring antenna and smart gear; b practical smart gear health monitoring system

Figure 2 denotes two experimental samples for the experimental works of this research and a laser sintering machine with conductive ink. Compared to the practical components of the smart gear system denoted in Fig. 1a, the two experimental samples in Fig. 2a are only two identical antennas. As mentioned in Sect. 2.1, the designs of the sintered patterns in Fig. 1a share a similar pattern of the antenna that has an open spiral coil and a GND part. This means the difference between the sintering patterns of the monitoring antenna and the smart gear is merely the sensor chain of the smart gear pattern. Consequently, the desired $Q_{2}$ and $\omega_{2}$ parameters of the sensor track circuit integrated on the smart gear side can be considered the parameters of the antenna circuit connected with the sensor track. That explains the necessity of the consideration of the case where two plain antennas are coupled initially in this study. Particularly, only the antenna circuit on the smart gear side is kept while the sensor track is removed. By doing this, it is able to clarify the roles of the integrated antenna in the sensor track circuit of the smart gear side in the magnetic coupling of the CM system. Moreover, the removal of the sensor chain can reduce the effect of the sensor part on the magnetic coupling of the system. One of our previous experimental works indicated that the coupling between the monitoring antenna and an uncracked sensor track circuit results in resonant return loss data that has numerous minor peaks on its chart instead of a smooth shape. The smooth shape of the mentioned resonant return loss data can be referred to from [12] with two smooth resonant peaks at the vicinity of 0.3 [GHz]. Apparently, the appearance of these minor peaks can take long time and effort of the data picking-up step in this research since the proposed method required picking up arbitrary points from the experimental data of the resonant return loss of the monitoring antenna. Therefore, the consideration of a sensor antenna coupled with a monitoring antenna is projected to be done in the next step of this study. Here, the substrate material of these two antenna samples in Fig. 2a is polyoxymethylene (POM). Due to the fact that the POM plastic gear is used typically in our laboratory, this POM plastic is selected as a substrate material for the monitoring antenna. To enhance the surface quality of the POM plate, one layer of polyimide tape is applied to cover the POM plate surface. The length and thickness of this square-shaped POM plate are 52 [mm] and 2 [mm]. The polyimide layer has a thickness of about 60 [µm]. Figure 2b and c show a laser sintering machine and a silver nanoparticle conductive ink NPS-J manufactured by Harima Kasei. The specification of this machine is provided in [13].

Fig. 2

Experimental samples and laser sintering technique: a two experimental samples of antennas on POM plates, b self-developed laser sintering printer, c silver nanoparticle NPS-J conductive ink

The process of fabrication of these practical samples on POM plates is referred to [13]. As mentioned, the POM plate is covered by one layer of polyimide tape for the improvement of the surface quality. After this taping step, the POM plate with the polyimide layer is treated in a vacuum plasma device to enhance the conductive quality of the sintered layer afterward. The time of this surface treatment is about 2 min. Next, the conductive ink NPS-J is sprayed directly on the surface of the polyimide taped POM plate via a spraying gun. To enhance the uniformity of the sprayed conductive ink layer, this POM plate is placed in a spin coater for spinning within 1 min at 1000 [rpm]. Then, the wet-ink layer on the polyimide taped POM plate is dried on a hot plate for about 10 min at 60 Celsius degrees. The laser sintering process is conducted by the 4-axis laser sintering machine in Fig. 2b. At the end of this laser sintering process, the surface of the polyimide taped POM plate has both the sintered layer of sensor and the residual conductive ink that has not been sintered. To obtain the experimental sample as Fig. 2a, acetone is used to remove all the residual conductive ink on the surface of the POM plate. In addition, RF connectors are attached to the two antenna samples by a special conductive glue. These connectors are useful for the connections between the antennas and any external devices or RF modules. Specifically, in the case of a monitoring antenna, the connector is connected to the network analyzer to get the return loss signal of this antenna.

Fundamentally, the experimental configuration of this research is set in accordance with the results of [12]. The experimental results obtained by [12] demonstrated that the physical conditions of the sensor track circuit on the smart gear side had a remarkable effect on the magnetic coupling between this sensor track circuit and the antenna circuit of the monitoring antenna. Moreover, according to the Sect. 2.1, the return loss signal of the monitoring antenna becomes a resonant return loss signal due to the magnetic coupling. In this way, any changes in the magnetic coupling result in the change in this resonant return loss signal.

2.2 Proposed methodology

In this research, the parameters of $Q_{2}$ and $\omega_{2}$ of the antenna sample on the smart gear side denoted on the right side in Fig. 2a are the desired. Therefore, a parameter identification method is proposed to identify the values of these two parameters. Figure 3 provides an outline of the content of this paper including the procedure of the proposed parameter identification method. As can be seen from this figure, a magnetic coupling diagram that describes how to allow the magnetic coupling to occur is provided. Since the two antenna models in Fig. 2a are similar, here and after, the monitoring antenna—on the left side is called the “1st antenna” and the smart gear side antenna—on the right side is called the “2nd antenna”. According to the magnetic coupling diagram, during the experiments, the 1st antenna is kept connected to the network analyzer while the 2nd antenna is connected to each of 3 external RF modules. These 3 RF modules are Open, Short, and 50 [Ω] modules, respectively. Basically, the connection of the 2nd antenna to these expressed 3 RF modules can change the characteristics or the conductive conditions of the 2nd antenna circuit into 3 cases designated as Open, Short, and Loaded by a 50 [Ω] resistor, respectively. In this way, the magnetic coupling of the 1st antenna and the three states of Open, Short, and Loaded by a 50 [Ω] resistor of the 2nd antenna result in three corresponding resonant return loss values of the 1st antenna. Thanks to the network analyzer, these resonant return loss values can be attained as the experimental data of this research.

Fig. 3

Magnetic coupling diagram and flow chart of research activity

In addition, to establish the return loss equation, an LCRs circuit is also proposed to express the magnetic coupling phenomenon between the 1st and 2nd antennas in Fig. 3. This LCRs circuit contains both the inductance, capacitance, and resistance components of the 1st antenna—the monitoring antenna and the 2nd antenna—the smart gear side antenna. Based on this LCRs circuit and Kirchhoff’s voltage law, a return loss equation has been derived. Unfortunately, this equation has more than one unknown variable, and the targeted $Q_{2}$ and $\omega_{2}$ values are just two of the variables of this equation. Hence, to solve this problem, the authors provide a technique to reduce the untargeted and unknown parameters. This technique focuses on using all available data that can be collected from the experimental data or the measurable parameters of the practical 1st antenna. Thanks to this technique, the return loss equation becomes an equation with only two unknown variables that are the targeted $Q_{2}$ and $\omega_{2}$ values. As a result, an equation system that includes two return loss equations can be established for the targeted $Q_{2}$ and $\omega_{2}$ values—the only two unknown values of this equation system. Since there are 3 different experimental data sets, there are 3 separate equation systems have been established, correspondingly. Therefore, the results of these 3 separate equation systems are 3 pairs of $Q_{2}$ and $\omega_{2}$ values. Finally, these obtained 3 pairs of $Q_{2}$ and $\omega_{2}$ values are validated via simulation works.

2.2.1 Experimental setting and data

This section interprets the experimental works of this research. The experimental setting in this research can be referred to from the coupling system schematic in Fig. 3. Specifically, two antennas 1st and 2nd are contacted and fixed on their backsides. The 1st antenna—the monitoring antenna remains connected to the network analyzer while the 2nd antenna is connected to 3 external RF modules as introduced in the previous section. Figure 4a denotes these RF modules that are the Open, Short, and 50 [Ω] modules, respectively. These modules belong to a precise SMA 4pcs CAL KIT that is tested and specialized for any RF testing application. Figure 4b denotes how an RF module is connected to the 2nd antenna via its connector. The network analyzer used in this study to observe and capture the resonant return loss data of the smart gear system is an Anritsu Vector Network Analyzer (VNA) with model MS46122B.

Fig. 4

External RF modules: a SMA CAL KIT with Adapter (1), Short module (2), and 50 [Ω] module (3); b 2nd antenna circuit in connection with external modules

Regarding the use of the RF modules to change the characteristics of the 2nd antenna, this comes from the constitution of the sensor track circuit. In fact, the sensor track circuit consists of an antenna pattern and a conductive sensor track. Here, the role of the conductive sensor track on the smart gear is to connect two poles of the antenna pattern which are the open spiral coil and the GND part. This connection aims to form a closed conductive sensor track circuit. As mentioned above, since the sensor track of the 2nd antenna is not considered in this research, it is replaced by the RF external modules, the Open, Short, and 50 [Ω] modules. Particularly, the connection between the 2nd antenna and the Open module means that the open spiral coil and the GND part are open. In other words, it is similar to the case where the sensor track is broken and non-conductive. On contrary, the connection between this 2nd antenna and the Short module means that the sensor track is uncracked and conductive. In the case of the 50 [Ω], it aims to mitigate the conductivity of the sensor track by adding about 50 [Ω] of resistance value to the total resistance value of the sensor track circuit. Therefore, the magnetic coupling between a similar monitoring antenna—the 1st antenna and the 2nd antenna attached correspondingly by each of 3 RF modules results in 3 different resonant return loss data. They are also called 3 different experimental results denoted in Fig. 8.

2.2.2 Parameter identification of ${Q}_{2}$ and ${\omega }_{2}$

In this section, the establishment of the resonant return loss equation system is interpreted. Moreover, the parameter identification technique is also discussed and solved.

2.2.2.1 Schematic LCR circuits

As discussed earlier, the magnetic coupling between the 1st antenna and the 2nd antenna help the authors indicate the physical conditions of the 2nd antenna wirelessly. This phenomenon motivates the authors to consider proposing an LCRs circuits system for this coupling system. Figure 5 denotes a proposed LCRs circuit. According to this figure, the proposed LCRs circuit has two loops of the LCRs circuit. Specifically, the loop on the left side represents the equivalent circuit of the 1st antenna while the loop on the right side represents the equivalent circuit of the 2nd antenna. The major components of the left-side loop of the LCRs circuit consist of $R_{1}$, $L_{1}$, $C_{1}$ that are the resistance, inductance, and capacitance of the 1st antenna’s circuit. Turning to the right-side loop,$R_{2}$,$L_{2}$,$C_{2}$ are the respective parameters of the 2nd antenna’s circuit. The other parameters include the $e_{s}$, $Z_{0}$, $I_{1}$, $I_{2}$, and $L_{12}$. Here, the values of $e_{s}$, $Z_{0}$ represent the power supply and its characteristic impedance. The $I_{1}$, $I_{2}$ represent the currents flowing in each closed loop of the LCRs circuit. And the $L_{12}$ represents the mutual inductance of the two antenna coils in a magnetically coupled system.

Fig. 5

LCRs circuit of two antennas coupling system

Theoretically, the values of $Q_{2}$ and $\omega_{2}$ or $Q$ and $\omega$ in general play important roles in a magnetically coupled system of antenna circuits. Specifically, the quality ($Q$) factor of an antenna circuit system is well-known as a common and straightforward value to evaluate both the bandwidth and the range of frequencies over which the antenna can operate correctly. The $Q$ parameter can be defined as the quotient value of the power stored in the reactive field and the radiated power. This $Q$ parameter is frequently measured by considering the number of times the current can pass through the circuit before the energy is radiated. By the way of explanation, this dimensionless factor—the $Q$ parameter can indicate the radiation efficiency of the antenna. In an antenna circuit, the resistance value is a major factor that can affect the $Q$ factor’s value significantly due to its obstacle to the current flow. As the consequence of the change of the $Q$ factor’s value, the natural angular frequency of the describing antenna circuit is also affected according to the relation of these two values, expressed via Eq. (4). Therefore, the values of $Q_{2}$ and $\omega_{2}$ of the 2nd LCRs loop in Fig. 5 must be targeted initially for the parameter identification problem.

2.2.2.2 The return loss equation

In order to establish the return loss equation, the authors apply the theory of Kirchhoff’s voltage law to the LCRs circuit in Fig. 5. Consequently, the following expressions can be written:

$$\left[ {\begin{array}{*{20}c} {R_{1} + j\omega L_{1} + \frac{1}{{j\omega C_{1} }}} & { - j\omega L_{12} } \\ { - j\omega L_{12} } & {R_{2} + j\omega L_{2} + \frac{1}{{j\omega C_{2} }}} \\ \end{array} } \right]\left[ {\frac{{I_{1} }}{{I_{2} }}} \right] = \left[ {\begin{array}{*{20}c} {e_{s} } \\ 0 \\ \end{array} } \right]$$

(1)

In this equation system, the total impedance values of each LCRs loop component are shown. Particularly, the $R_{1} + j\omega L_{1} + 1/j\omega C_{1}$ represents the total impedance value of the 1st antenna’s circuit and the $R_{2} + j\omega L_{2} + 1/j\omega C_{2}$ represents the total impedance value of the 2nd antenna’s circuit. Moreover, the $- j\omega L_{12}$ value represents the effect of the mutual inductance value $L_{12}$ of the magnetically coupled system with $\omega$ is the resonant angular frequency. By dividing both sides of the component equations in the equation system (1) by the inductance values $L_{1}$ and $L_{2}$, respectively, the equation system (1) becomes:

$$\left[ {\begin{array}{*{20}c} {\frac{{R_{1} }}{{L_{1} }} + j\omega + \frac{1}{{j\omega L_{1} C_{1} }}} & { - j\omega \frac{{L_{12} }}{{L_{1} }}} \\ { - j\omega \frac{{L_{12} }}{{L_{2} }}} & {\frac{{R_{2} }}{{L_{2} }} + j\omega + \frac{1}{{j\omega L_{2} C_{2} }}} \\ \end{array} } \right]\left[ {\frac{{I_{1} }}{{I_{2} }}} \right] = \left[ {\begin{array}{*{20}c} {\frac{{e_{s} }}{{L_{1} }}} \\ 0 \\ \end{array} } \right]$$

(2)

In reference to the theory of the LCRs circuit, the basic formulae of the angular frequency and the $Q$ value are defined as:

$$\omega = \frac{1}{{\sqrt {LC} }}$$

(3)

And

$$Q = \frac{\sqrt L }{{R\sqrt C }} = \frac{\omega L}{R} = \frac{1}{R\omega C}$$

(4)

Next, by substituting the fundamental formulae in (3) and (4) into the expression system (2), the expression system (2) turns to:

$$\left[ {\begin{array}{*{20}c} {\frac{{\omega_{1} }}{{Q_{1} }} + j\omega + \frac{{\omega_{1}^{2} }}{j\omega }} & { - j\omega \frac{{L_{12} }}{{L_{1} }}} \\ { - j\omega \frac{{L_{12} }}{{L_{2} }}} & {\frac{{\omega_{2} }}{{Q_{2} }} + j\omega + \frac{{\omega_{2}^{2} }}{j\omega }} \\ \end{array} } \right]\left[ {\frac{{I_{1} }}{{I_{2} }}} \right] = \left[ {\begin{array}{*{20}c} {\frac{{e_{s} }}{{L_{1} }}} \\ 0 \\ \end{array} } \right]$$

(5)

From the equation system (5), the expression representing the value of $I_{1} /e_{s}$ can be deduced as:

$$\frac{{I_{1} }}{{e_{s} }} = \frac{1}{{L_{1} }}\frac{{\frac{{\omega_{2} }}{{Q_{2} }} + j\omega + \frac{{\omega_{2}^{2} }}{j\omega }}}{{\left( {\frac{{\omega_{1} }}{{Q_{1} }} + j\omega + \frac{{\omega_{1}^{2} }}{j\omega }} \right)\left( {\frac{{\omega_{2} }}{{Q_{2} }} + j\omega + \frac{{\omega_{2}^{2} }}{j\omega }} \right) + \omega^{2} \frac{{L_{12}^{2} }}{{L_{1} L_{2} }}}}$$

(6)

Moreover, according to the definition of the return loss, it is stated that the return loss value is the ratio of the reflected power and the incident power. Thus, in this study, the return loss of the 1st antenna – the monitoring antenna in Fig. 5, can be expressed as follows:

$$R.L = 10\log_{10} \left| {\frac{{P_{1r} }}{{P_{1i} }}} \right|$$

(7)

where $P_{1r}$ is the power of the reflected wave and $P_{1i}$ is the power of the incident wave, respectively. In theory, $P_{1r} = \left| {V_{1r} } \right|\left| {I_{1r} } \right|$ and $P_{1i} = \left| {V_{1i} } \right|\left| {I_{1i} } \right|$ with $V_{1i} + V_{1r} = V_{in}$ and $I_{1i} - I_{1r} = I_{in}$ where $V_{1i}$, $V_{1r}$, $V_{in}$ and $I_{1i}$, $I_{1r}$, $I_{in}$ are voltages and currents of the incident wave, the reflected wave, and the input power source, correspondingly. From these expressions, the reflected coefficients of voltage and the current can be derived and transformed as: $\Gamma_{V} = V_{1r} /V_{1i} = \left( {Z_{in} - Z_{0} } \right)/\left( {Z_{in} + Z_{0} } \right)$ and $\Gamma_{I} = - I_{1r} /I_{1i} = - \left( {Z_{in} - Z_{0} } \right)/\left( {Z_{in} + Z_{0} } \right)$ so the Eq. (7) becomes:

$$R.L = 10\log_{10} \left| {\frac{{Z_{in} - Z_{0} }}{{Z_{in} + Z_{0} }}} \right|^{2} = 20\log_{10} \left| {\frac{{Z_{in} - Z_{0} }}{{Z_{in} + Z_{0} }}} \right|$$

(8)

As referred to from chapter 7 (Sect. 7.1) in [19], the expressions of $I_{in} = I_{1}$ and $V_{in} = e_{s} - I_{1} Z_{0}$ with $Z_{0}$ is the characteristic impedance of the cable are obtained. So, the following Eq. (8) can be rewritten as a general return loss equation:

$$R.L = 20\log_{10} \left| {1 - 2Z_{0} \frac{{I_{1} }}{{e_{s} }}} \right|$$

(9)

Then, by combining the expression (6) with the general return loss Eq. (9), the generalized return loss equation of the coupling system in Fig. 5 is finally represented as:

$$R.L = 20\log_{10} \left| {1 - 2\frac{{Z_{0} }}{{L_{1} }}\frac{{\frac{{\omega_{2} }}{{Q_{2} }} + j\omega + \frac{{\omega_{2}^{2} }}{j\omega }}}{{\left( {\frac{{\omega_{1} }}{{Q_{1} }} + j\omega + \frac{{\omega_{1}^{2} }}{j\omega }} \right)\left( {\frac{{\omega_{2} }}{{Q_{2} }} + j\omega + \frac{{\omega_{2}^{2} }}{j\omega }} \right) + \omega^{2} \frac{{L_{12}^{2} }}{{L_{1} L_{2} }}}}} \right|$$

(10)

2.2.2.3 Determining the unknown parameters

Equation (10) signifies the relation between the components of the LCRs circuit in Fig. 5 and the return loss values. As is shown, although Eq. (10) contains both the targeted $Q_{2}$ and $\omega_{2}$ values, it also contains a number of unknown parameters. These unknown parameters are the characteristic impedance $Z_{0}$, the $Q_{1}$ value, the natural angular frequency $\omega_{1}$, the resistance $R_{1}$, the inductance $L_{1}$ and $L_{2}$, the return loss value $R.L.$, the resonant angular frequencies $\omega$, and the mutual inductance $L_{12}$. Unfortunately, due to the existence of these unknown values, the target of obtaining directly the $Q_{2}$ and $\omega_{2}$ values from this equation becomes impossible. Hence, in order to resolve this problem, it is a must to reduce the number of these unknown parameters of the Eq. (10). In this way, the Eq. (10) can become an equation of only $Q_{2}$ and $\omega_{2}$ values.

The characteristic impedance $Z_{0}$: In the beginning, the characteristic impedance $Z_{0}$ is focused. Since the value of $Z_{0}$ is basically determined by the geometrical shape and material of the transmission line of wave propagation, it is typically provided by the manufacturers of the adapter, connector, cable, and network analyzer. In this research, an Anritsu VNA model MS46122B and its accessories are used to attain the experimental data. So, in reference to the specification of this device and its manufacturer’s information, the value of this $Z_{0}$ parameter is decided as 50 $\left[ \Omega \right]$. This 50 $\left[ \Omega \right]$ is also the default characteristic impedance of almost all high-frequency devices.

The natural angular frequency $\omega_{1}$: Next, according to the relation between the return loss value and the eigenfrequency of an antenna provided in [14], the natural angular frequency $\omega_{1}$ of the 1st antenna can be estimated based on the return loss value of this antenna at a single mode—without any magnetic coupling with any other antenna or smart gear. This return loss value is measured directly through the network analyzer by connecting only the 1st antenna to one port of the network analyzer. Based on the chart of the return loss value of the 1st antenna at single mode, the antenna’s natural frequency is selected as the frequency value of the first peak. In fact, our experimental work also reveals that this first peak is the only peak that will change noticeably when the magnetic coupling of the system occurs. According to Fig. 9, the first peak of the return loss signal of the 1st antenna at single mode is the peak that has a frequency value of about 0.30 [GHz]. Therefore, the natural frequency of the single 1st antenna is selected as $f_{1} \cong$ 0.30 [GHz] and the value of natural angular frequency can be calculated as $\omega_{1} = 2\pi f_{1}$[rad/s].

The inductance value $L_{1}$: Regarding the inductance value $L_{1}$ of the 1st antenna in Eq. (10), studies by [15] and [16] proposed that the inductance $L_{1}$ of a spiral coil of an antenna can be estimated based on the following Eqs. (11) and (12). Since the coil of the 1st antenna in this research is a spiral coil, the Eqs. (11) and (12) can be used to compute the inductance $L_{1}$ value of this antenna. Figure 6 shows the detached parts and specifications of the inner spiral coil of the 1st antenna in Fig. 2a—the monitoring antenna including the called straight part and the spiral part. Here, the Eq. (11) is applied for a spiral part and the Eq. (12) is applied for a straight part of the spiral coil.

$$L_{spr} = \frac{{\mu_{0} n^{2} d_{avg} c_{1} }}{2}\left( {\ln \left( {\frac{{c_{2} }}{\rho }} \right) + c_{3} \rho + c_{4} \rho^{2} } \right)$$

(11)

$$L_{str} = \frac{{\mu_{0} }}{{6\pi a^{2} }}\left( {3a^{2} l\ln \left( {\frac{{l + \sqrt {l^{2} + a^{2} } }}{a}} \right) - \left( {l^{2} + a^{2} } \right)^{\frac{3}{2}} + 3al^{2} \ln \left( {\frac{{a + \sqrt {a^{2} + l^{2} } }}{l}} \right) + l^{3} + a^{3} } \right)$$

(12)

where $\mu_{0}$ is the magnetic permeability of the vacuum environment,$n$ is the number of spiral turns, and $d_{in}$,$d_{out}$ are the inner and outer diameters of the spiral part.$d_{avg}$ is the averaged diameter of the spiral part that can be calculated as $d_{avg} = 1/2\left( {d_{in} + d_{out} } \right)$. Then, the value of the fill ratio $\rho = \left( {d_{in} - d_{out} } \right)/\left( {d_{in} + d_{out} } \right)$ can be obtained. $l$ and $a$ represent the length and width of the straight part also indicated in Fig. 6. Moreover, the values of the coefficients for current sheet expression $c_{i}$ with $i = 1,...,4$ are available in [15] with “Circle” layout. In total, the parameters for calculating the inductance value $L_{1}$ are collected and provided in Table 1. As a result of Eqs. (11) and (12), the inductance of the monitoring antenna is obtained as $L_{1} = L_{spr} + L_{str} \cong$$1.73 \times 10^{ - 7}$[H].

Fig. 6

Spiral part and straight part of the spiral antenna coil

Table 1 Parameters for the calculation of the inductance $L_{1}$

The resistance value $R_{1}$: Another unknown parameter that is able to estimate via the practical model of the 1st antenna is the resistance value $R_{1}$. This value is measured directly from the sample of the 1st antenna using an electric tester device. Specifically, the value of $R_{1}$ is equal to the resistance value measured between two endpoints of the open spiral coil. In this way, the resistance value of the antenna coil in the 1st antenna is achieved as $R_{1} \cong$ 133 $\left[ \Omega \right]$. So far, the $Q_{1}$ value becomes the final unknown value of the 1st antenna in the Eq. (10). Luckily, this $Q_{1}$ value can be estimated via the fundamental expression (4) thanks to the obtained values of the natural angular frequency $\omega_{1}$, the inductance $L_{1}$, and this measured resistance value $R_{1}$. Consequently, the $Q_{1}$ value of the 1st antenna can be computed and obtained as $Q_{1} \cong$$2.30$.

Up to now, the remaining unknown values are just the values of the 2nd antenna. The values comprise the natural angular frequency $\omega_{2}$, the inductance $L_{2}$, and the value of $Q_{2}$. Besides, the mutual inductance $L_{12}$ of the magnetic coupling of two antennas has also not been defined yet. Due to the fact that these parameters are unable to be gauged simultaneously during the coupling process, they cannot be estimated in the same way used for the 1st antenna. However, since the physical condition of the 2nd antenna remains intact during the magnetic coupling process, it is assumed that the physical characteristics of this antenna also remain intact. This means the inductance $L_{2}$ value of this 2nd antenna becomes constant. Moreover, the experimental setting describes that the two POM plate antennas are fixed to each other on their backsides. In this way, the coupled distance of the two 1st and 2nd antennas is constant during the magnetic coupling process. Thus, the mutual inductance $L_{12}$ that occurs between two open spiral coils of the two antennas can be considered a constant value. As a result, the $Q_{2}$ and $\omega_{2}$ values of the smart gear side antenna become the major variables of the generalized return loss Eq. (10) that have the most significant impact on the change in the resonant return loss of the magnetically coupled system. In other words, the derived generalized return loss Eq. (10) has been turned into a two-variable equation of the $Q_{2}$ and $\omega_{2}$ values.

The coupling coefficient $k$: In addition, the reference of [17,18,19], and [20] of the authors suggests that the reactance value of the resonant circuits system must be equal to 0 $\left[ \Omega \right]$ due to the occurrence of the magnetic coupling phenomenon. Typically, unlike resistors, the value of reactance in a resonant LCRs circuit system depends only on the frequency and the existence of resistance that resists currents in the closed loop without dissipating power. Hence, to reach the maximum value of the currents in the circuit, the inductive reactance must be equal to the capacitive reactance, so the value of the general reactance becomes 0 $\left[ \Omega \right]$. Moreover, the existence of the resistance value $R_{1}$ and $R_{2}$ in the left and right LCRs loop circuits in Fig. 5 majorly plays a role in the consumed loads or the radiation losses whereas the other inductance values $L_{1}$, $L_{2}$, and capacitance values $C_{1}$, $C_{2}$ are the essential components contributing to the magnetic coupling process. That is to say, the resistance values have less effect on the magnetic coupling process. So, for the sake of simplicity, in combination with the condition of the general reactance value in the resonant circuit system, the resistance values, therefore, can be ignored. By this assumption, the LCRs loops on the left and right sides in Fig. 5 can become the ideal LC circuit loops for the magnetic coupling process. As referred to from the resonant condition explained via the Magnetic Coupling section of Chapter 7, Eq. (7.56) in [19], the admittance equation of the lumped-element circuit model in Fig. 5 can also be written as:

$$\frac{1}{{\omega L_{12} }} + \frac{1}{{\omega \left( {L_{1} - L_{12} } \right) - \frac{1}{{\omega C_{1} }}}} + \frac{1}{{\omega \left( {L_{2} - L_{12} } \right) - \frac{1}{{\omega C_{2} }}}} = 0$$

(13)

Furthermore, since the design and specification of the 1st antenna and the 2nd antenna are identical, the inductance and capacitance values of these two antennas can be assumed as $L = L_{1} = L_{2}$,$C = C_{1} = C_{2}$ during the magnetic coupling process. Then, the following two resonant angular frequencies can be satisfied:

$$\omega_{m} = \frac{\omega }{{\sqrt {1 + k} }} = \frac{1}{{\sqrt {\left( {L + L_{12} } \right)C} }}$$

(14)

$$\omega_{e} = \frac{\omega }{{\sqrt {1 - k} }} = \frac{1}{{\sqrt {\left( {L - L_{12} } \right)C} }}$$

(15)

Here, the two expressions (14) and (15) are also agreed with the Eq. (7.39) and (7.40) in [19]. On one hand, in the expressions (14) and (15),$\omega_{m}$ and $\omega_{e}$ are known as two angular frequency values of two resonant peaks of the magnetically coupled system’s signal. In this research, the two resonant peaks can be seen in the chart of the resonant return loss signal of the 1st antenna – the return loss signal of the 1st antenna measured when the magnetic coupling is occurring between the antenna 1st and 2nd. Hence, these $\omega_{m}$ and $\omega_{e}$ values are estimated directly from this resonant return loss signal—the experimental data. On the other hand, as discussed earlier, the values of $\omega_{m}$ and $\omega_{e}$ can be obtained only when the resonance happens in the coupling system. It means that the selection of $f_{m}$ and $f_{e}$ for calculating these $\omega_{m}$ and $\omega_{e}$ values must be away from the two existing fixed nodes which are commonly found in a resonant signal chart. Figure 10 denotes an example of selecting the values of $f_{m}$ and $f_{e}$ from the experimental data of the magnetic coupling between the 1st antenna and the 2nd antenna that is at a Short state. Similarly, the values of $f_{m}$ and $f_{e}$ for the Open and Loaded by a 50 $\left[ \Omega \right]$ resistor case can be obtained. Table 2 provides the selected values of $f_{m}$ and $f_{e}$ for all three cases.

Table 2 Selected values from the experimental data

According to the expressions (14) and (15), the value of $k$ can be estimated via the following expression:

$$k = \frac{{\omega_{e}^{2} - \omega_{m}^{2} }}{{\omega_{e}^{2} + \omega_{m}^{2} }} = \frac{{f_{e}^{2} - f_{m}^{2} }}{{f_{e}^{2} + f_{m}^{2} }}$$

(16)

In this Eq. (16), the value of $k$ is called the coupling coefficient of the magnetically coupled system. As referred to the picked resonant frequencies $f_{e}$ and $f_{m}$ in Table 2, the computed values of $k$ for the Open, Short, and Loaded by a 50 [Ω] resistor states are $k_{open} \cong$ 0.27,$k_{short} \cong$ 0.29, and $k_{50Ohms} \cong$ 0.32, respectively. Conceptually, the value of the coupling coefficient $k$ represents the proportion of the coupled magnetic flux between two coils in a common magnetic flux. This common magnetic flux is produced by an electric current flowing in one coil of a magnetically coupled system. In the authors’ developing smart gear system, the monitoring antenna or the 1st antenna in this research is kept connected to the network analyzer during the magnetic coupling. Thus, there must be a current $I_{1}$ flowing in the open spiral coil of this 1st antenna with a high frequency value. Consequently, this current $I_{1}$ enables the open spiral coil to produce its magnetic flux. In this way, the open spiral coil of the 2nd antenna – the smart gear side antenna will become a passive coil that is influenced by the magnetic flux generated by the 1st antenna. Figure 7 shows the magnetic fluxes generated by the spiral coils of the 1st and 2nd antennas. Particularly,$\vec{\user2{B}}_{1}$ and $\vec{\user2{B}}_{2}$ are vectors of the magnetic fluxes generated by the current $I_{1}$ in coil 1 and current $I_{2}$ in coil 2, respectively. As a result, the change of the current $I_{1}$ in coil 1 involves the change of the magnetic flux in coil 2 and vice versa.

Fig. 7

Relation between the current $I_{i}$ and the magnetic flux

The mutual inductance $L_{12}$: Likewise, the mutual inductance $L_{12}$, which is known as a fundamental value in an electromagnetic induction device such as an electric transformer or motor, is also related to the magnetic flux between these two spiral coils. Specifically, the value of the mutual inductance presents the magnetically coupled proportion of two coils in a common magnetic flux. Unlike the coupling coefficient $k$, the value of mutual inductance $L_{12}$ helps indicate how much the voltages can be induced in an adjacent coil of the magnetic flux produced coil. Consequently, to indicate the coupled characteristics of a magnetic coupling system, it is critical to analyze and establish the relation between the coupling coefficient k and this mutual inductance value $L_{12}$.

In reference to Fig. 7, due to the flowing of the current $I_{1}$ in the open spiral coil of the 1st antenna on the left or the 1st spiral coil, assumed that $\Phi_{1}$ is the magnetic flux generated by this 1st spiral coil whereas $\Phi_{12}$ is the coupled magnetic flux value between the 1st and the 2nd spiral coils of the 1st and 2nd antennas, then the value of coupling coefficient $k_{1}$ can be computed as follows:

$$k_{1} = \frac{{\Phi_{12} }}{{\Phi_{1} }}$$

(17)

In accordance with [21] and [22], the self-inductance value $L_{1}$ of this coil can be obtained as:

$$L_{1} = \frac{{N_{1} \Phi_{1} }}{{I_{1} }}$$

(18)

The mutual inductance value induced by the 1st spiral coil on the 2nd coil $L_{{12_{1} }}$ can be computed:

$$L_{{12_{1} }} = \frac{{N_{2} \Phi_{12} }}{{I_{1} }}$$

(19)

In the expression (18) and (19),$N_{1}$ and $N_{2}$ are the number of turns of the 1st spiral coil and the 2nd spiral coil, respectively.

Similarly, due to the influence of the magnetic flux of the 1st coil on the 2nd coil, there will be a current $I_{2}$ in the 2nd spiral coil. In this case,$\Phi_{2}$ is the magnetic flux produced by the 2nd spiral coil and $\Phi_{21}$ is the coupled magnetic flux value between the 2nd and the 1st spiral coil and $\Phi_{2}$ is the magnetic flux produced by the 2nd spiral coil. Here, the value of $k_{2}$ is able to be calculated as follows:

$$k_{2} = \frac{{\Phi_{21} }}{{\Phi_{2} }}$$

(20)

The self-inductance value $L_{2}$:

$$L_{2} = \frac{{N_{2} \Phi_{2} }}{{I_{2} }}$$

(21)

And the mutual inductance value induced by the 2nd coil on the 1st coil $L_{{12_{2} }}$:

$$L_{{12_{2} }} = \frac{{N_{1} \Phi_{21} }}{{I_{2} }}$$

(22)

Furthermore, since the coupling conditions of the system and the distance between two spiral coils are unchanged during the magnetic coupling process, the coupling coefficient $k$ and the mutual inductance value $L_{12}$ must be constant. This means $k_{1} = k_{2} = k$ and $L_{{12_{1} }} = L_{{12_{2} }} = L_{12}$. From (17); (19) and (20); (22), the following expressions can be obtained:

$$L_{12} = \frac{{N_{2} \Phi_{1} k}}{{I_{1} }}$$

(23)

And

$$L_{12} = \frac{{N_{1} \Phi_{2} k}}{{I_{2} }}$$

(24)

By multiplying the expression (23) by (24) side-by-side and then combining it with the expression (18) and (21), the relation between the mutual inductance $L_{12}$ and the coupling coefficient $k$ can be achieved as:

$$L_{12}^{2} = k^{2} L_{1} L_{2}$$

(25)

Thanks to the expression (25), the generalized return loss Eq. (10) is able to be rewritten as:

$$R.L. = 20\log_{10} \left| {1 - 2\frac{{Z_{0} }}{{L_{1} }}\frac{{\frac{{\omega_{2} }}{{Q_{2} }} + j\omega + \frac{{\omega_{2}^{2} }}{j\omega }}}{{\left( {\frac{{\omega_{1} }}{{Q_{1} }} + j\omega + \frac{{\omega_{1}^{2} }}{j\omega }} \right)\left( {\frac{{\omega_{2} }}{{Q_{2} }} + j\omega + \frac{{\omega_{2}^{2} }}{j\omega }} \right) + \omega^{2} k^{2} }}} \right|$$

(26)

2.2.2.4 Identification of Q ₂ and ω ₂

The identification technique for $Q_{2}$ and $\omega_{2}$ values is expressed in this part. With regards to the return loss Eq. (26), it is apparent to realize that although the unknown parameters consisting of $Q_{1}$, $\omega_{1}$, $L_{1}$, and $k$ in this function have been estimated and calculated, it is still impossible to solve this equation. The reason comes from the values of the return loss $R.L.$ and the resonant angular frequency $\omega$. These two values of a magnetically coupled system remain unrevealed. Nevertheless, as stated in the introduction section, this research proposes a method to identify the parameters of the practical smart gear samples. It means that the experimental data must be used for the parameter identification process. In the scope of this research, the practical smart gear samples are the two 1st and 2nd antennas. The experimental data of these two antennas are denoted in Fig. 8. In scrutiny of this figure, the experimental data show that both the return loss and frequency values are available. Fortunately, as explained above, the data in Fig. 8 are the resonant return loss of the 1st antenna measured during the magnetic coupling of the 1st and 2nd antenna. In this way, the use of the resonant return loss and the resonant angular frequency values of the experimental data of these two 1st and 2nd antennas can help reduce another two remaining unknown values—the return loss $R.L.$ and the resonant angular frequency $\omega$ in Eq. (26).

Fig. 8

Experimental return loss data of the smart gear system

Moreover, the experimental data contains a number of feature points. Each of these feature points has its own practical return loss and resonant frequency values. Then, when an arbitrary feature point is selected and replaced for the unknown $R.L.$ and $\omega$ in the Eq. (26), a two-variable equation of the unknown $Q_{2}$ and $\omega_{2}$ can be obtained. In this research, since the targeted unknown parameters include both the $Q_{2}$ and $\omega_{2}$ values, an equation system of two two-variable equations is required for the calculation. Said differently, two arbitrary points are essentially picked up from the experimental data of each case in three total cases including the Open, Short, and Loaded by a 50 [Ω] resistor. These two points have their specific return loss $R.L.A_{i}$(with i = 1,2) and resonant frequency $f_{Ai}$ or resonant angular frequency $\omega_{Ai}$. By replacing these corresponding values for the unrevealed $R.L.$ and $\omega$ values in Eq. (26), the two Eqs. (27) and (28) are obtained. The combination of these two equations forms an equation system of two two-variable equations or a two-dimensional simultaneous equation system. The solutions of this equation system are the value of $Q_{2}$ and $\omega_{2}$ of the 2nd antenna’s circuit—the antenna on the smart gear side.

$$R.L.A_{1} = 20\log_{10} \left| {1 - 2\frac{{Z_{0} }}{{L_{1} }}\frac{{\frac{{\omega_{2} }}{{Q_{2} }} + j\omega_{A1} + \frac{{\omega_{2}^{2} }}{{j\omega_{A1} }}}}{{\left( {\frac{{\omega_{1} }}{{Q_{1} }} + j\omega_{A1} + \frac{{\omega_{1}^{2} }}{{j\omega_{A1} }}} \right)\left( {\frac{{\omega_{2} }}{{Q_{2} }} + j\omega_{A1} + \frac{{\omega_{2}^{2} }}{{j\omega_{A1} }}} \right) + \omega_{A1}^{2} k^{2} }}} \right|$$

(27)

$$R.L.A_{2} = 20\log_{10} \left| {1 - 2\frac{{Z_{0} }}{{L_{1} }}\frac{{\frac{{\omega_{2} }}{{Q_{2} }} + j\omega_{A2} + \frac{{\omega_{2}^{2} }}{{j\omega_{A2} }}}}{{\left( {\frac{{\omega_{1} }}{{Q_{1} }} + j\omega_{A2} + \frac{{\omega_{1}^{2} }}{{j\omega_{A2} }}} \right)\left( {\frac{{\omega_{2} }}{{Q_{2} }} + j\omega_{A2} + \frac{{\omega_{2}^{2} }}{{j\omega_{A2} }}} \right) + \omega_{A2}^{2} k^{2} }}} \right|$$

(28)

Incidentally, to attain more accurate solution values of $Q_{2}$ and $\omega_{2}$ from the established equation system, the technique of selecting the two arbitrary points from the experimental data is also pivotal. Typically, similar to the selection of $f_{m}$ and $f_{e}$ in the previous section, the desired arbitrary points should be the points that are influenced significantly by any change occurring to the magnetic coupling between the two antennas. Based on the authors’ experience, the points that have their resonant frequency in the vicinity of the resonant peak that has the lowest absolute resonant return loss value should be selected. Specifically, Fig. 11 illustrates an example of the selection for two arbitrary points from the experimental data of the Open case. Similarly, the other values of the arbitrary points selected for the Short and Loaded by a 50 [Ω] resistor cases can be selected. Table 2 provides all the selected arbitrary points for all three cases.

By substituting the estimated values of $\omega_{1}$, $R_{1}$, $k_{open}$, $k_{short}$, $k_{50Ohms}$,$Q_{1}$ and the selected values of the arbitrary points in Table 2 into both Eqs. (27) and (28), three equation systems corresponding with the Open, Short, and Loaded by a 50 [Ω] resistor states of the 2nd antenna—the smart gear side antenna are accomplished. Then, these equation systems are solved in a MATLAB environment. The solutions are pairs of $Q_{2}$ and $\omega_{2}$ values.

To validate the integrity of the solutions, the MATLAB environment is also used to calculate the simulation data. This calculation is based on the return loss Eq. (26) in Sect. 2.2.2.3. Basically, the calculation for the simulation data is the inverse of the return loss Eq. (26) where the values of $R.L.$ and $\omega$ are unknown instead of the $Q_{2}$ and $\omega_{2}$ values. The other measured and estimated values consist of $\omega_{1}$, $R_{1}$, $k$ including $k_{open}$, $k_{short}$, and $k_{50Ohms}$ for all 3 cases, and $Q_{1}$ have already remained intact. In this way, the validated simulation data are the data that have resonant peaks and the best match with the original experimental data of their corresponding cases. Obviously, the most valuable $Q_{2}$ and $\omega_{2}$ values are the values that produce the validated simulation data.

3 Results and discussion

This section shows the results of the research. The results include the experimental results and the simulation results. Besides, the obtained results are also analyzed and discussed.

3.1 Results

Figure 8 denotes the experimental data used in this research. These data are measured directly from the 1st antenna thanks to the network analyzer. In this figure, the experimental data includes both the return loss data of the 1st antenna at single mode and the resonant return loss data of this 1st antenna measured during the magnetic coupling process. The magenta solid line represents the return loss of the 1st antenna in “Single Mode”. The other red, green, and blue solid lines represent the resonant return loss signal of the 1st antenna when this antenna is magnetically coupled with the 2nd antenna at different states. Specifically, the red color, the green color, and the blue color represent the resonant return loss data measured when the 2nd antenna is in the Open state—the connector is left open, Short state—the connector is connected to the RF Short module, and Loaded by a 50 [Ω] resistor state—the connector is connected to the RF 50 [Ω] module, respectively.

According to this Fig. 8, it is simple to realize the difference between the return loss data and the resonant return loss data of the 1st antenna. This difference comes from the number of peaks on the data charts. Particularly, in the range of 0 [GHz] to 0.6 [GHz], especially at 0.3 [GHz], the experimental data of the 1st antenna when this 1st antenna is at single mode has only one peak while the other three experimental data when this 1st antenna is at magnetic coupling mode have all two peaks. Moreover, in accordance with three different states of the 2nd antenna when its connector is left Open, connected to the RF Short and 50 [Ω] module, the shapes of the resonant return loss data of the 1st antenna are also separate for each state.

Figures 9, 10, and 11 are examples of using the experimental data for the reduction of unknown parameters step in the previous Sect. 2.2.2.3. Figure 9 denotes the estimation of the natural frequency f₁ of the 1st antenna via its return loss data measured at a single mode. Figure 10 shows an example of selecting the values of $f_{m}$ and $f_{e}$ from the resonant return loss data of the magnetic coupling system. Here, the connector of the 2nd antenna is connected to the RF Short module when this 2nd antenna is magnetically coupled with the 1st antenna. Figure 11 provides a visual example of picking up two arbitrary points from the resonant return loss data in the case where the connector of the 2nd antenna is left open.

Fig. 9

Estimation of the natural frequency f₁ of the 1st antenna—the monitoring antenna

Fig. 10

Example of the selection of $f_{m}$ and $f_{e}$ from the experimental data of the Short case

Fig. 11

Example of the selection of two arbitrary points from the experimental data of the Open case

Turning to the values of $Q_{2}$ and $\omega_{2}$, the solutions of the MATLAB solving process are the numerous pairs of $Q_{2}$ and $\omega_{2}$ provided in Tables 3, 4, and 5 in corresponding to three cases of the 2nd antenna during the magnetic coupling with the 1st antenna. These three cases are Open, Short, and Loaded by a 50 [Ω] resistor states, respectively. Certainly, the pairs of $Q_{2}$ and $\omega_{2}$ values that are equal to 0 in these three tables are invalid and discarded. In reference to the expression in Sect. 2.2.2.4, the calculated values of $Q_{2}$ and $\omega_{2}$ in three Tables 3, 4, and 5 are replaced back to the Eq. (26) to compute their respective simulation data. After this step, the comparison of fitness between the computed simulation data and the original experimental data for each case of the physical condition of the 2nd antenna helps the authors to select the most valuable $Q_{2}$ and $\omega_{2}$ values. Figures 12, 14, and 16 denote the simulation results of all calculated values of $Q_{2}$ and $\omega_{2}$ for the Open, Short, and Loaded by a 50 [Ω] resistor states of the 2nd antenna, respectively.

Table 3 Solutions of $Q_{2}$ and $\omega_{2}$ in the case of Open state

Table 4 Solutions of $Q_{2}$ and $\omega_{2}$ in the case of Short state

Table 5 Solutions of $Q_{2}$ and $\omega_{2}$ in the case of Load by a 50 $\left[ \Omega \right]$ resistor state

Fig. 12

Comparison of the simulation results and the experimental data for the Open case

In scrutiny to Fig. 12, the red solid line represents the experimental data of the magnetic coupling between the 1st antenna and the 2nd antenna in the case where its connector of the 2nd antenna is left Open. Besides, there are four simulation results represented by green, cyan, purple, and blue solid lines, respectively. These simulation results are calculated using the identified $Q_{2}$ and $\omega_{2}$ values in Table 3. Although the number of pairs of $Q_{2}$ and $\omega_{2}$ values is eight pairs, only four pairs are used for computing the simulation results. This is caused by the coincidence of the solution pairs and the invalid zero-value solutions. The four considered pairs of $Q_{2}$ and $\omega_{2}$ values that are (1.86; 1.816 × 10⁹ [Rad/s]), (3.67; 2.06 × 10⁹ [Rad/s]), (0.55; 2.24 × 10⁹ [Rad/s]), and (2.96; 2.32 × 10⁹ [Rad/s]). In an overview of the four simulation results and the experimental data, the two-peak shape of the chart of the experimental data shows that the 1st and 3rd results of the simulation work are not suitable and removed since both have only one resonant peak on their charts’ shapes. In addition, the 2nd and the 4th simulation results show that both have two resonant peaks on their charts. However, the 4th simulation result certainly has the best fitness with the experimental data compared to the rest of the simulation results. Based on this finding, the pair of $Q_{2}$ and $\omega_{2}$ values used to produce the 4th simulation result in Fig. 12 are recognized as the most valuable $Q_{2}$ and $\omega_{2}$ values of the 2nd antenna in this Open case. The final values for this case of $Q_{2}$ and $\omega_{2}$ are 2.96 and 2.32 × 10⁹ [Rad/s].

Figure 13 illustrates only the simulation result of the selected identified $Q_{2}$ and $\omega_{2}$ values and the original experimental data of the magnetically coupled system where the 2nd antenna is in the Open state. In this figure, the fitness between the simulation result—presented in blue color and the experimental data—presented in red color can be seen clearly.

Fig. 13

Comparison of the simulation result of the best identified parameters and the experimental data for the Open case

Similarly, in the second case where the physical condition of the 2nd antenna is in a Short state, the identified values of $Q_{2}$ and $\omega_{2}$ are also specified in Table 4.

From this table, the four pairs of identified $Q_{2}$ and $\omega_{2}$ values are (1.32; 1.717 × 10⁹ [Rad/s]), (2.74; 1.9 × 10⁹ [Rad/s]), (1.42; 2.06 × 10⁹ [Rad/s]), and (3.73; 1.854 × 10⁹ [Rad/s]). In corresponding to Fig. 14, the simulation results calculated using these four pairs of $Q_{2}$ and $\omega_{2}$ values are presented by solid lines in green, cyan, purple, and blue colors. In comparison to the other three solid lines, the blue solid line of the 4th simulation result shows that it matches the red color line—the experimental data highly with minimum fitness values between the two results. In other words, the final identified values of $Q_{2}$ and $\omega_{2}$ values of this Short state of the 2nd antenna are obtained with $Q_{2}$ = 3.73 and $\omega_{2}$ = 1.854 × 10⁹ [Rad/s]. Figure 15 exemplifies the best fitness of the 4th simulation result and the experimental data in detail.

Fig. 14

Comparison of the simulation results and the experimental data for the Short case

Fig. 15

Comparison of the simulation result of the best identified parameters and the experimental data for the Short case

Regarding the third case where the 2nd antenna is Loaded by a 50 [Ω] resistor, the results achieved in Table 5 indicate that there are only 3 potential pairs of $Q_{2}$ and $\omega_{2}$ values instead of 4 pairs like the previous two cases are suitable for the validation process. In this way, the simulation results of these three pairs of $Q_{2}$ and $\omega_{2}$ values that are (1.13; 2.05 × 10⁹ [Rad/s]), (1.89; 1.843 × 10⁹ [Rad/s]), and (1.30; 2.16 × 10⁹ [Rad/s]) are denoted in Fig. 16. In this figure, the best fitness between the 2nd simulation result and the experimental result proves that only the pair of $Q_{2}$ and $\omega_{2}$ values that produce the 2nd simulation result are suitable for the selection. In this pair, the values of $Q_{2}$ and $\omega_{2}$ are 1.89 and 1.843 × 10⁹ [Rad/s], respectively. Figure 17 denotes the best fitness of the 2nd simulation result of the selected $Q_{2}$ and $\omega_{2}$ values and the original experimental data of the Loaded by a 50 [Ω] resistor case.

Fig. 16

Comparison of the simulation results and the experimental data for the Loaded by a 50 $\left[ \Omega \right]$ resistor case

Fig. 17

Comparison of the simulation result of the best identified parameters and experimental data for the Loaded by a 50 [Ω] resistor case

The total final 6 identified values of $Q_{2}$ and $\omega_{2}$ for all three cases are summarized in Table 6.

Table 6 Final identified $Q_{2}$ and $\omega_{2}$ values for 3 cases of characteristics

3.2 Discussion

By observing Fig. 8, at first glance, there is a significant difference and transformation between the forms of all the return loss and resonant return loss data of the 1st antenna—the monitoring antenna. Particularly, in the vicinity of 0.3 [GHz], the one-peak return loss form of the 1st antenna in “Single Mode” denoted in the magenta color has been transformed to the two-peak return loss form when the 2nd antenna is brought close to its backside. Here, the term “Single Mode” describes the state of the 1st antenna when it is connected to the network analyzer without coupling with any other antenna or smart gear. In addition, although the change of the resonant return loss signals’ shapes between the Short and Loaded by a 50 [Ω] resistor states of the 2nd antenna is considerable, the comparison of these two signals’ shapes with the resonant signal shape of the Open state shows a great transformation. As discussed in Sect. 2.2.1, the Open state of the 2nd antenna is equivalent to the unhealthy or cracked condition of the conductive sensor track. Hence, these findings affirm that the selection of 3 RF modules to change the characteristics of the magnetic coupling between the 1st antenna and the 2nd antenna is acceptable. Subsequently, the RF external modules can replace the roles of the conductive sensor track on the 2nd antenna in this research. Moreover, the transformation of the one-peak return loss form into the two-peak resonant return loss form of the 1st antenna due to the appearance of the 2nd antenna also consolidates and supports the background of our developing gear health monitoring system. That is to say, the condition of the gear can be monitored wirelessly via the return loss of the monitoring antenna.

Figures 9, 10, and 11 demonstrate the application of the results of a practical experiment in resolving a theoretical problem. As mentioned in Sect. 2.2.2.3, the first peak’s frequency of the return loss data measured from the 1st antenna when it is in the single mode in Fig. 9 is selected as the natural frequency of the 1st antenna. This selection is based on the fact that with different conditions of the magnetic coupling, the first peak’s frequency is the frequency value where the shape of the resonant return loss signal is merely observed to change in accordance with each condition of the magnetic coupling. The typical change is from a one-peak shape to a two-peak shape and vice versa. In the case of Fig. 10, this figure denotes an example of the selection of the two resonant frequency values $f_{m}$ and $f_{e}$ of the Short case. Here, the main point of this selection technique for all cases is that the values of $f_{m}$ and $f_{e}$ must be in the vicinities of the two resonant peaks but different from the two fixed frequency values. Since these two fixed frequency values are known as unchanged even under the condition of the magnetic coupling change, the selection of $f_{m}$ and $f_{e}$ that coincides with these frequency values results in unreliable values of the coupling coefficient $k$ and the other components of the return loss equation. Figure 11 shows another example of the section of two arbitrary points from the experimental data for the calculation of the return loss Eq. (26) in the Open case. As stated earlier, the two arbitrary points are selected mainly in the vicinity of the second peak instead of the first peak. This comes from the behavior of the two peaks of resonant return loss data where the first peak is often considered an unstable peak compared to the second peak.

In fact, the LCRs circuit and the return loss equation are theoretically proposed and derived. So, the use of the experimental results can not only reduce the number of the unknown variables of this equation but also make the derived equation become more practical and accurate. As introduced previously, the resonant return loss data of the 1st antenna—the monitoring antenna in our smart gear system is the unique output data. And this data can provide the unknown values of $R.L.$ and $\omega$ in the Eq. (26) for the equation solving step. However, in the scope of this research, instead of only the values of $R.L.$ and $\omega$, the addition of the values such as the natural frequency f₁ or the inductance value $L_{1}$ selected directly from the practical samples that produce the described output data to the return loss Eq. (26) can be more advantageous for the parameter identification process and the final values of $Q_{2}$ and $\omega_{2}$ can be more practical values.

In addition, the selected data in Figs. 13, 15, and 17, on one hand, is an evidence that the fitness between the simulation result and the experimental data tends to reduce when the frequency value reaches the middle range of 0.4 [GHz] and 0.5 [GHz], typically around 0.45 [GHz], and afterward. This is considered the result of the energy loss in the practical antenna model compared to a computed ideal mathematical model. Typically, our developing gear health monitoring method is based on the first peak of the resonant return loss value received by the monitoring antenna. This first peak is mostly located in the vicinity of 0.3 [GHz] of the resonant frequency axis for the current designed dimensions of the antenna pattern. Besides, the frequency range within 0.45 [GHz] of the return loss signal chart is also the range where the magnetic coupling will occur. In other words, compared to the frequency range from 0.5 [GHz] and afterward, the frequency range within 0.45 [GHz] of the return loss signal chart becomes more fruitful for not only the visual monitoring but also the characteristic parameter identification process of the 2nd antenna’s physical conditions. On the other hand, the inspection of the data presented in Figs. 13, 15, and 17 indicates that the fitness between the two types of data in Fig. 13—the result of the Open case has the best quality compared to the other Figs. 15 and 17- the results of the Short case and the Loaded by a 50 [Ω] resistor case, respectively. This means the parameter identification method applied to identify the $Q_{2}$ and $\omega_{2}$ values of the 2nd antenna in the Open case works more effectively than for the other Short and Loaded by a 50 [Ω] resistor cases. In a real working condition, the finding of the cracked or unhealthy states—the Open state of the 2nd antenna is always much more substantial than in the other two cases—the Short and Loaded by a 50 [Ω] resistor cases.

Regarding Table 6, there are several noticeable points. At the first glance, the final identified values of $Q_{2}$ and $\omega_{2}$ are different for each case of Open state, Short State, and Loaded by a 50 [Ω] resistor state. Fortunately, this finding proves that the change in the physical condition of the 2nd antenna has a significant impact on the $Q_{2}$ and $\omega_{2}$ values of this 2nd antenna’s circuit. Next, in the case where the 2nd antenna is in the Open state, the change of its natural angular frequency $\omega_{2}$ or the natural frequency $f_{2}$ is remarkable compared to the similar values of the other Short state and Loaded by a 50 [Ω] resistor state cases. This finding can be considered the most beneficial point for the condition monitoring of gears in our developing smart gear system. Specifically, the change of the natural angular frequency, in this Open case, induces a clear change of the resonant return loss signal form. According to this point, any damages that occur to gears, therefore, can be recognized more accurately. Moreover, the comparison of the Short case and the Loaded by a 50 [Ω] resistor case also indicates that although the values of $Q_{2}$ have changed considerably between the two cases, the values of $\omega_{2}$ are almost unchanged. From this point, the authors can state that the variation of the resistance values in an LCRs circuit results mostly in the variation of the $Q$ factor values. Here, the values of $Q_{2}$ have been decreased when a 50 [Ω]resistor is added to the LCRs circuit. This finding agrees with the expression of the relation between the $Q$ value and the inductance, capacitance, and resistance values since the $Q$ value is inversely proportional to the resistance value in a circuit. Therefore, monitoring the $Q_{2}$ factor values of the 2nd antenna—the antenna on the smart gear side during the gear operation is valuable to the detection of the smart gear’s physical condition prior to the appearance of severe cracks.

4 Conclusion

In our developing smart gear monitoring system, parameter identification is considered an indispensable part of the technique. In this research, the authors have considered 3 magnetic coupling conditions in accordance with three conditions of the 2nd antenna—the smart gear side antenna in this research. Since the 2nd antenna is attached respectively to each of the 3 RF external modules to change its characteristic conditions, the three conditions are called Open, Short, and Loaded by a 50 [Ω] resistor, correspondingly. A parameter identification method is proposed to identify the characteristic parameters of the 2nd antenna for all three cases of its different characteristic conditions. The targeted values of the parameter identification method are the $Q_{2}$ and $\omega_{2}$ values of the 2nd antenna. The achievements of the works in this manuscript can be listed below:

• The proposed method has indicated that it is successful in identifying the values of $Q_{2}$ and $\omega_{2}$ values of the 2nd antenna in accordance with each of the three characteristic conditions of this antenna. The input data for the parameter identification is only the experimental data measured when the 1st antenna—the monitoring antenna is magnetically coupled with the 2nd antenna in all three cases. This means the values of $Q_{2}$ and $\omega_{2}$ values of the 2nd antenna have been identified wirelessly via the output data of the 1st antenna.

• A return loss function has been accomplished. This function contains the vital parameters of a resonant LCRs circuit system such as the natural angular frequency $\omega_{1}$ or the mutual inductance $L_{12}$. Hence, the derived function is not only useful for the parameter identification purpose in this research but also useful for computing applications for other magnetic coupling systems such as a wireless power transfer system or an antenna system.

• The validation of the identified $Q_{2}$ and $\omega_{2}$ values based on the original experimental data proves the propriety of the identified values. The validation results indicate that the identified parameters are reliable and helpful for indicating the specific conditions of the practical models. Specifically, the identified values of $Q_{2}$ and $\omega_{2}$ for each of the Open, Short, and Loaded by a 50 [Ω] resistor cases are different. Besides, the changing tendencies of the $Q_{2}$ or $\omega_{2}$ values in each of the three cases such as the Short case and the Loaded by a 50 [Ω] resistor case are also different. These differences contribute to the consolidation of the proposed parameter identification method.

• The findings about the relation between the identified characteristic values—the $Q_{2}$ and $\omega_{2}$ values and the conditions of the sensor chain (or a conductive track replaced by the RF modules in this research) of the 2nd antenna—the smart gear side antenna show that they are relevant to each other, especially in the Open case. This case is equivalent to the cracked condition of the sensor chain on the smart gear side.

Nevertheless, with the exception of the inspected Open case, the analysis of the identified values of $Q_{2}$ and $\omega_{2}$ of the Short state case and the Loaded by a 50 [Ω] resistor case has indicated a drawback. To be specific, the error gaps between the identified simulation data and the experimental data of the two mentioned cases are much higher than the error gap in the Open state case. Basically, since the return loss equation contains numerous unknown parameters including the $Q_{2}$ and $\omega_{2}$ values, the identification of only the $Q_{2}$ and $\omega_{2}$ values have shown that it is inadequate to afford a higher accurate condition monitoring result. Hence, shortly, the authors’ target will focus on the improvement of the currently proposed parameter identification method. In this way, the factors that can influence the magnetic resonant return loss of the monitoring antenna must be considered more. Then, the more characteristic parameters of the smart gear side antenna, including the $Q_{2}$ and $\omega_{2}$ values, can be identified.