Modeling and Transmission Characteristics Study of a Resonant Underwater Wireless Electric Power Transmission System

Abstract

Compared to the traditional wet-mate underwater power supply method, Magnetic Coupling Resonant Wireless Power Transfer (MCR-WPT) technology boasts advantages such as excellent insulation, high safety, and convenient operation, showing promising application prospects in the field of power supply for underwater vehicles and other mobile underwater devices. In order to explore the transmission characteristics of this technology underwater, this article first establishes a traditional mathematical model, and then modifies the underwater model through analysis of changes in coil self-inductance and mutual inductance, as well as the impact of eddy current losses. Using the modified mathematical model of the underwater MCR-WPT system, the transmission characteristics are analyzed, and simulations and experimental validations are performed using MATLAB R2022a software. In the study of frequency characteristics, it is found that the system operates optimally when both ends of the circuit work at the resonant state; that is, when f_input = f_resonance = 100 kHz, the output performance is at its best, and the optimal resonant frequency significantly improves power and transmission efficiency. When the input frequency is less than 87.3 kHz or greater than 122.9 kHz, the output power decreases to less than half of the maximum power. In the investigation of load effects, the optimal load for maximizing system output power was identified, but the load that maximizes transmission efficiency is different from this optimal load. This study provides strong theoretical support and guidance for improving the performance of underwater wireless power transmission systems.

1. Introduction

Underwater mobile devices play a crucial role in the exploration of marine resources, and the stability and safety of their energy supply have become focal points in research fields [1]. Traditional energy replenishment methods, including manual recovery and wet-mate wired charging, rely on electrical contact to provide energy, necessitating periodic replacement and maintenance. Manual recovery reduces the operational flexibility of underwater devices, and when carrying out long-term underwater missions, the demand for energy forces the use of larger batteries, which limits the efficiency of underwater operations. At the same time, the process of manual recovery lowers the level of automation and affects stealth. On the other hand, wet-mate charging demands high sealing requirements, is complex to operate, and wear and tear over time during the operation may shorten the lifespan of the equipment [2,3]. Wireless power transfer technology adopts a non-contact approach to achieve energy transmission from the transmitter to the receiver, avoiding safety hazards such as spark generation and leakage that could arise from traditional power supply methods, thereby enhancing the stability and safety of the energy supply. In the unique environment of underwater, wireless power transfer technology demonstrates its unique advantages [4].

The resonant underwater wireless electric power transmission system is a new type of electric power transmission technology based on the principle of magnetic coupling resonance [5]. Its basic principle involves establishing resonant magnetic coupling in an underwater environment to transmit electric power from the transmitter to the receiver. Compared with traditional wireless electric power transmission technologies, the resonant system offers higher transmission efficiency and distance [6], thus holding great potential in underwater environments.

In 1994, researchers from the University of Wisconsin, including B.J. Heeres, developed an underwater wireless electric power transmission system with a coupling device that achieved up to 85% transmission efficiency using coaxial windings, and the apparent power reached the 3 kVA level [7]. In 2004, Japan Electric Corporation (NEC) and a research team from Northeastern University collaborated to develop a wireless charging system for AUVs, which used ferrite magnetic cores and conical coils to adapt to ocean current disturbances, achieving a maximum output power of 500 W and an efficiency of 96% [8]. In 2014, research by Awai at Yamaguchi University in Japan on MCR-WPT in seawater found that double-layer helical coils effectively reduced eddy current losses [9,10]. In 2019, a research team from Tokyo University of Marine Science and Technology developed a new coil layout to optimize AUV wireless charging performance, with a charging platform that supports reliable electric power transmission for any landing position, using a double-layer multi-coil structure, and achieved a transmission efficiency of over 74% [11]. Zhejiang University has developed a wireless power transmission system using a tank-type magnetic coupling structure for underwater equipment. Its working gap is 8 mm, the transmission power is 500 W, and the underwater wireless charging efficiency can reach 75% [12]. In addition, the team also adopted a conical magnetic coupling structure to construct a 300 W UWPT system [13]. Zhang Kehan’s team from Northwestern Polytechnical University proposed a multi-directional underwater wireless charging concept, with multiple transmitters and receivers in the transmission device, ensuring stable mutual inductance coefficients unaffected by rotation or separation [14,15,16]. In reference [17], the eddy current density distribution of the coupling mechanism cross-section in air, fresh water, and seawater was simulated, and the changes in magnetic core loss, winding loss, and additional eddy current loss with current and frequency were simulated in these three media. Reference [18] designed a wireless loosely coupled resonant induction system for a 50 kg manned submersible, using a permanent magnet core coupler. When the gap distance varies between 6 and 10 mm, the system transmits about 500 W of power, and the efficiency can reach 88%. The team from Harbin Institute of Technology is committed to the magnetic coupling design and underwater energy stability control research of AUV underwater wireless charging systems, and has proposed a ε-type magnetic coupling device and arc-shaped coil structure to enable the maximum transmission power of the arc-shaped coil to reach 600 W [19]. Jacek Maciej Stankiewicz from Bialystok University of Technology analyzed a wireless power transmission system with multiple resonators and designed experiments to verify the calculation results [20].

In the field of MCR-WPT technology, extensive experimental research has been conducted, achieving significant results. The research focuses mainly on optimizing coil structure layout and system integration to reduce transmission losses in seawater, as well as the theoretical analysis of offset issues and factors affecting losses in underwater transmission. However, most studies discuss magnetic coupling underwater transmission from an electromagnetic field perspective, with insufficient theoretical research on model establishment and modification of MCR-WPT systems in seawater and transmission characteristics. The main research problem addressed in this study is the inefficiency of existing underwater power transmission methods. In response to the shortcomings of existing research, this article aims to explore the underwater transmission characteristics of the system and achieve efficient transmission of resonant underwater wireless power transmission systems through better parameter matching and better transmission characteristics. Specifically, this paper analyzes the principles of the MCR-WPT system, establishes a mathematical model, analyzes changes in underwater coil self-inductance and mutual inductance, and modifies the model. It then analyzes the system’s transmission characteristics and performs simulations and experimental validations using MATLAB and Ansys Maxwell software, providing guidance for improving the performance of underwater wireless electric power transmission systems. By modeling the system and studying its transmission characteristics, we can provide a theoretical basis and technical support for the development of resonant underwater wireless power transmission systems and promote their widespread adoption in practical applications. The research in this field not only has important scientific value but also has broad application prospects including providing new solutions and ideas for the energy supply of underwater mobile devices, addressing the unique challenges faced by underwater wireless energy transmission, and promoting exploration and development in the deep-sea field.

2. System Analysis and Modeling

2.1. System Operating Principle

The MCR-WPT system generally comprises two coils that are coupled through a high-frequency electromagnetic field generated at the transmitting end. Both ends of the system are designed to maintain a consistent resonant frequency. When the receiving coil couples with the high-frequency electromagnetic field, it gradually reaches a resonant state. In this state, the inductive reactance and capacitive susceptance present in the circuit are significantly attenuated, resulting in the impedance behaving almost resistively. Consequently, energy losses in the inductors and capacitors can be largely ignored [21]. Figure 1 illustrates the structural diagram of this system.

The transmission end must be connected to the power grid to provide energy supply for the entire system; hence, it is an active circuit. The energy supplied by the grid is a low-frequency alternating current, which is converted into a high-frequency alternating current through various stages of circuit processing before being transmitted to the transmitting coil [22,23]. Under the influence of currents, the transmitting coil in the dielectric space excites a high-frequency alternating magnetic field, thus enabling the receiving coil to couple and receive energy. When establishing the mathematical model, the circuit preceding the transmitting coil is generally equivalent to a high-frequency AC power source.

2.2. Mathematical Model Construction

In the MCR-WPT system, the load often exhibits purely resistive, capacitive, or inductive characteristics. Since these types of loads have a similar impact on the overall system, this paper will focus solely on the purely resistive load. To simplify the calculations, the circuit adopts a series–series compensation structure. The equivalent circuit model of the MCR-WPT system is depicted in Figure 2.

In this model, the transmitting end is denoted by the subscript $p$, and the receiving end is denoted by the subscript $s$. Except for $R_L$ as the load, the other two resistors represent the equivalent internal resistance of the coil.

Assuming ω is the system angular frequency, the equivalent impedances at the transmitting and receiving ends are expressed as follows:

(1)$\left\{\begin{array}{l}{Z}_p=R_p+j\omega L_p+\frac{1}{j\omega C_p}\\ Z_s=R_s+R_L+j\omega L_s+\frac{1}{j\omega C_s}\end{array}\right.$

According to Kirchhoff’s voltage law, the currents can be calculated as:

(2)$\left\{\begin{array}{l}{i}_p=\frac{Z_s{U}_S}{Z_p{Z}_s+{\left(\omega M\right)}^2}\\ i_s=\frac{j\omega M{U}_S}{Z_p{Z}_s+{\left(\omega M\right)}^2}\end{array}\right.$

Let the input power of the transmitting end of the system be $P_{in}$, and the output power of the receiving end load be $P_{out}$, as can be obtained from Formulas (1) and (2):

(3)$\left\{\begin{array}{c}{P}_{in}=U_S{i}_p=\frac{Z_s{U}_s^2}{Z_p{Z}_s+{\left(\omega M\right)}^2}\hfill \\ P_{out}=R_L{i}_S=\frac{{\left(\omega M\right)}^2{U}_s^2{R}_L}{{\left[Z_p{Z}_s+{\left(\omega M\right)}^2\right]}^2}\end{array}\right.$

Consequently, the overall system transmission efficiency can be deduced to be:

(4)$\eta =\frac{(\omega M{)}^2{R}_L}{Z_s[Z_p{Z}_s+(\omega M{)}^2]}\times 100\%$

For ease of circuit design at both the transmitting and receiving ends, the parameters of the transmitting and receiving coils should be kept consistent, with the coil’s self-inductance and internal resistance values being equal, that is, $L_p=L_s=L, R_p=R_s=R$. When the equivalent system model operates in a resonant state, the circuit behaves resistively, that is, $Z_p=R_p, Z_s=R_s+R_L$, at which point the circuit impedance is at its minimum and the current is at its maximum, the transmission efficiency is:

(5)$\eta =\frac{(\omega M{)}^2{R}_L}{(R_s+R_L)\left[R_p\right(R_s+R_L)+(\omega M{)}^2]}\times 100\%$

As can be seen from Equation (5), the system’s transmission efficiency is related to the load resistance, resonant frequency, mutual inductance, and the internal resistance of both the transmitting and receiving coils. Since the resonant frequency, mutual inductance, and internal resistance are related to the resonant coil, it is crucial for the entire wireless power transmission system.

2.3. Variations in Self-Inductance and Mutual Inductance of Underwater Coils

The underwater environment causes changes in the characteristics of the coupling mechanism of the MCR-WPT system, necessitating an applicability study of the general mutual inductance model in air.

2.3.1. Variations in Self-Inductance of Underwater Coils

The two most commonly applied coil structures are the planar spiral coil and the cylindrical spiral coil. Under the same conditions, compared to the cylindrical spiral coil, the planar spiral coil is more suitable. The coil’s own fundamental parameter is self-inductance, and the most important parameter when two coils are coupled is mutual inductance. These two parameters are crucial for the system. The self-inductance can be ascertained using the following empirical formula; however, it is important to note that this formula is applicable exclusively to coils operating at frequencies below 30 MHz and only when the condition $W>2R$ is met [24].

(6)$L=\frac{N^2}{25.4}\frac{R^2}{\left(8R+11W\right)}$

The coil’s number of turns is denoted as $N$, where $R$ represents its average radius, and $W$ corresponds to the difference between the coil’s maximum outer diameter and its minimum inner diameter.

The reactance of each turn of the coil is [25]:

(7)$X_{\mathrm{sea}}=\omega \mu a{N}^2\left[K(k)-2-\frac{\pi }{3}(\beta a{)}^3+\frac{4}{15}(\beta a{)}^4-\cdots \right]$

The coil’s radius is denoted by a, where $K\left(k\right)$ denotes the complete elliptic integral of the first kind:

(8)$K(k)={{\displaystyle \int }}_0^{\frac{\pi }{2}}\frac{d\varphi }{{(1-k^{2}si{n}^2\varphi )}^{1/2}}$

One can express the reactance of a coil with a single turn in air as follows:

(9)$X_{\mathrm{a}\mathrm{i}\mathrm{r}}=\omega \mu a{N}^2\left[K\right(k)-2]$

By substituting the reactance values from air to seawater, one can establish the connection between the impedance characteristics of a single-turn coil in these two different environments as follows:

(10)${\mathrm{X}}_{\mathrm{sea}}=X_{\mathrm{air}}-\omega \mu a{N}^2\left[\frac{\pi }{3}(\beta a{)}^3-\frac{4}{15}(\beta a{)}^4-\cdots \right]\hspace{1em}$

In the Equation, $\beta =\sqrt{\omega \mu \sigma }$, where $\sigma$ denotes the conductivity of the conductive medium. The term subtracted on the right side of the equation, excluding $X_{\mathrm{air}}$, signifies the variation in reactance caused by the conductive medium, specifically seawater. Using MATLAB R2022a software, the variation of inductance in both air and seawater is plotted from 10 kHz to 10 MHz, as depicted in Figure 3. Based on the above analysis and the trend of the curve in the graph, it can be concluded that compared to air, the trend of inductance change in seawater is decreasing. However, in the frequency range of 10 kHz to 1 MHz, the difference between the two is very small [26].

2.3.2. Variations in Mutual Inductance of Underwater Coils

In the MCR-WPT system, the mutual inductance between coupling coils is also a significant factor affecting the system’s transmission characteristics. In the presence of a conductive transmission medium, high-frequency time-varying magnetic fields induce eddy currents within the medium. These currents weaken the induced electromagnetic field, thereby reducing the coupling efficiency between the two coils. Additionally, the conductive medium can interfere with the transmitter coil, influencing the phase of the magnetic field detected by the receiver coil. This interference affects the overall performance and reliability of the wireless transmission system.

Ansys Maxwell is used to perform finite element analysis on the system comprising the transmitter and receiver coils. Let the height of the medium be $h$ and its radius $R$, placed coaxially and symmetrically between the two coils, with both coils having a radius of $r$. The material of the conductive medium is set to air and seawater, and the magnetic field distribution is depicted as shown in Figure 4. Observing the variation of magnetic field vectors within the dashed rectangle reveals the following insights: In the absence of a conductive medium, as depicted on the left side of the diagram where the cylinder is filled with air (assumed to have zero conductivity), the magnetic field exhibits linear polarization. Conversely, in the presence of a conductive medium, such as seawater with non-zero conductivity, eddy currents are induced. These eddy currents interact with the original magnetic field, resulting in elliptical polarization of the magnetic field. This transformation signifies a decrease in the strength of the normal magnetic field between the coils, thereby impacting the coupling efficiency between them.

Seawater, relative to the coupling mechanism, can be considered an infinitely large medium. Assuming the ratio of the transmission medium’s radius to the coil radius is R/r, the curves depicting the variation of mutual inductance under different conductivity levels are illustrated in Figure 5. As it increases, the changes in both curves remain slight. However, when the conductivity of the medium reaches a certain threshold, the decreasing trend of the two curves becomes notably pronounced. This observation indicates that in seawater, traditional models show minimal alteration in mutual inductance as R/r increases.

Based on the analysis conducted, it is evident that across the frequency spectrum from 10 $\mathrm{k}\mathrm{H}\mathrm{z}$ to 10 $\mathrm{M}\mathrm{H}\mathrm{z}$, both self-inductance and mutual inductance values in seawater exhibit minimal deviation from those observed in air. This finding suggests that traditional models for self-inductance and mutual inductance remain valid and applicable in underwater environments.

2.4. Model Correction in Seawater

Seawater is rich in charged ions, leading to a reduction in the energy of high-frequency electromagnetic fields [27]. Thus, when compared to the equivalent circuit model in air, it is necessary to consider the eddy current loss characteristics specific to seawater. According to electromagnetic theory, the complex form of Maxwell’s equation is:

(11)$\left\{\begin{array}{l}\mathsf{\nabla }\times H=\sigma E+j\omega \epsilon E\hfill \\ \mathsf{\nabla }\times E=-j\omega \mu H\hfill \\ \mathsf{\nabla }\cdot H=0\hfill \\ \mathsf{\nabla }\cdot E=0\hfill \end{array}\right.$

Analyzing a sinusoidal wave propagating along the Z-axis in the positive direction and applying Maxwell’s equations allows for deriving the electric field solution of the wave. This analysis facilitates the formulation of an expression for the average power density applicable to diverse media:

(12)$\overline{S}=\frac{1}{2}{R}_e\left[e_x{E}_{xm}{e}^{-\alpha z}{e}^{-j\beta z}{e}_y\frac{E_{xm}}{\left|{\eta }_c\right|}{e}^{-\alpha z}{e}^{j\left(\beta z+\varphi \right)}\right]=\frac{1}{2}\frac{E_{xm}^2}{\left|{\eta }_c\right|}{e}_z{e}^{-2\alpha z}\cos \varphi$

Here, $r(r=\alpha +j\beta )$ denotes the propagation constant of the medium, where $\alpha$ represents the attenuation constant and $\beta$ denotes the phase constant; $E_{xm}$ signifies the initial electric field strength of the propagating electromagnetic field within the medium; ${\eta }_c$ denotes the intrinsic impedance of the medium; $\varphi$ represents the phase angle associated with the wave propagation.

Due to its non-zero conductivity, seawater causes attenuation of electromagnetic waves; hence, the attenuation constant $a\ne 0$, indicating that plane waves will experience energy loss while propagating through seawater. The conductivity of seawater at room temperature is over a thousand times greater than that of freshwater, with a value of $\sigma =4 \mathrm{s}/\mathrm{m}$ under the operating frequency of 100 kHz, the phase constant is $\beta \approx 1.26 \mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{m}$, and the attenuation constant is approximately $a\approx 1.26 \mathrm{N}\mathrm{p}/\mathrm{m}$. The intrinsic impedance of seawater is ${\eta }_c\approx 0.44e^{j45°}\Omega$. Substituting these values into the previous formula gives us the following result [28]:

(13)$\stackrel{-}{S}=\frac{1}{2}\frac{E_{xm}^2}{\left|{\eta }_c\right|}{e}_z{e}^{-2az}\cos \varphi =0.8E_{xm}^2{e}_z{e}^{-2.52}W/m^2$

(14)$\Delta =\frac{S_d}{S_0}=\frac{0.8E_{xm}^2{e}^{-2.52d}}{0.8E_{xm}^2}=e^{-2.52d}$

After adjusting the transmission characteristic indicators for systems in seawater, the analysis indicates that as the system operates within seawater, the energy demonstrates exponential decay over increasing transmission distances. If the distance is too great, the energy transmitted to the receiver can be almost negligible, resulting in extremely poor transmission performance of the system. From Equations (2) and (9), the input power $P_{in}$ and the output power $P_{out}$ can be determined as follows:

(15)$P_{in}=\frac{U_S^2{Z}_s}{Z_p{Z}_s+{\omega }^2{M}^2}$

(16)$P_{out}=\frac{U_S^2{\omega }^2{M}^2{R}_L}{{\left(Z_p{Z}_s+{\omega }^2{M}^2\right)}^2}{e}^{-2.52d}$

According to $P_{in}$ and $P_{out}$, the transmission efficiency of the system is:

(17)$\eta =\frac{{\omega }^2{M}^2{R}_L}{\left(Z_p{Z}_s+{\omega }^2{M}^2\right)Z_D}{e}^{-2.52d}$

When the system is in resonance, both ends of the circuit exhibit resistive behavior. The quality factor of the transmitting circuit is denoted as $Q_p=\omega L_p/Z_p=\omega L_p/R_p$, and that of the receiving circuit as $Q_s=\omega L_s/Z_s=\omega L_s/{(R}_s+R_L)$; the coupling coefficient between the two coils is expressed as $k=M/\sqrt{L_p{L}_s}$. Substituting the above parameters yields:

(18)$P_{out}=\frac{U_S^2{k}^2{Q}_p{Q}_s{R}_L}{{\left(k^2{Q}_p{Q}_s+1\right)}^2}{e}^{-2.52d}$

(19)$\eta =\frac{k^2{Q}_p{Q}_s{R}_L}{(k^2{Q}_p{Q}_s+1)Z_D}{e}^{-2.52d}$

The coupling coefficient between the coils varies with the effective transmission distance. The relationship between the transmission distance and the coils is given by:

(20)$d={\left(\frac{{\mu }_0\pi N_p{N}_s{r}_p^2{r}_s^2}{2k\sqrt{L_p{L}_s}}\right)}^{1/3}$

In the formula, ${\mu }_0=4\mathsf{\pi }\times {10}^{-7} \mathrm{H}/\mathrm{m}$, $N_p$ and $N_s$ are the number of turns in the transmitting and receiving end coils, $r_p$ and $r_s$ are the radius of the transmitting and receiving end coils, $k$ is the coupling coefficient, $L_p$ and $L_s$ are the inductance of the transmitting and receiving end coils, respectively.

When the distance is less than or equal to ${\left(\frac{{\mu }_0\pi N_p{N}_s{r}_p^2{r}_s^2}{2k\sqrt{L_p{L}_s}}\right)}^{1/3}$ and $k=1$, the system’s output power and transmission efficiency are articulated within Equations (18) and (19), respectively. When the distance is greater than ${\left(\frac{{\mu }_0\pi N_p{N}_s{r}_p^2{r}_s^2}{2k\sqrt{L_p{L}_s}}\right)}^{1/3}$ and $0\le k\le 1$, the following equations describe the output power and transmission efficiency:

(21)$P_{out}=\frac{U_S^2{\left(\frac{{\mu }_0\pi N_p{N}_s{r}_p^2{r}_s^2}{2d^3\sqrt{L_p{L}_s}}\right)}^2{Q}_p{Q}_s{R}_L}{{\left[{\left(\frac{{\mu }_0\pi N_p{N}_s{r}_p^2{r}_s^2}{2d^3\sqrt{L_p{L}_s}}\right)}^2{Q}_p{Q}_s+1\right]}^2}{e}^{-2.52d}$

(22)$\eta =\frac{{\left(\frac{{\mu }_0\pi N_p{N}_s{r}_p^2{r}_s^2}{2d^3\sqrt{L_p{L}_s}}\right)}^2{Q}_p{Q}_s{R}_L}{\left[{\left(\frac{{\mu }_0\pi N_p{N}_s{r}_p^2{r}_s^2}{2d^3\sqrt{L_p{L}_s}}\right)}^2{Q}_p{Q}_s+1\right]Z_D}{e}^{-2.52d}$

3. Magnetic Coupling Resonant Wireless Power Transfer System Transmission Characteristics

Output power and transmission efficiency are two main indicators of the transmission characteristics of a wireless power transfer system. From the previous analysis, it is known that the parameters affecting the output power and transmission efficiency include the resonant frequency, mutual inductance, load resistance, coil resistance, and power supply voltage, with the main parameters and their symbols shown in Table 1. The primary research focuses on how these influencing parameters affect the output power and transmission efficiency. Since the transmission distance between the two coils is very small, according to the revised mathematical model, it is known that the values of these two indicators are approximately equal to those of the traditional model, hence the impact of the coefficients is no longer considered.

3.1. Frequency Characteristics

3.1.1. Impact of Resonant Frequency

When the system operates in a resonant state, the frequency of the system is:

(23)$f=\frac{1}{2\pi }\sqrt{\frac{1}{L_p{C}_p}}=\frac{1}{2\pi }\sqrt{\frac{1}{L_s{C}_s}}$

The system’s efficiency is highest at the resonant frequency. Assuming that the circuit parameters at both ends remain consistent, the output power and system efficiency are as follows:

(24)$P_{out}=\frac{(2\pi fM{)}^2{U}_S^2{R}_L}{\left[R_p\right(R_s+R_L)+(2\pi fM{)}^2{]}^2}$

(25)$\eta =\frac{(2\pi fM{)}^2{R}_L}{(R_s+R_L)\left[R_p\right(R_s+R_L)+(2\pi fM{)}^2]}\times 100\%$

From Equation (23), it is evident that the system’s resonant frequency is related to mutual inductance and compensating capacitance. When these two parameters change, the resonant frequency also changes as per the calculation expression for the resonant frequency, indicating that the MCR-WPT system is influenced by multiple parameters.

With $R_p=R_s=0.55 \mathsf{\Omega }$, $M=17.44 \mathsf{\mu }\mathrm{H}$, $R_L=16 \mathsf{\Omega }$, and $U_S=20 \mathrm{V}$, the impact of resonant frequency on system characteristics in the MCR-WPT system is illustrated in Figure 6. As the designed resonant frequency of the system increases, the output power and transmission efficiency initially rise rapidly, reaching a peak before the output power sharply decreases while the transmission efficiency remains almost unchanged. When the system’s set resonant frequency increases to 66 kHz, the power is only half of its maximum value, and when the resonant frequency continues to increase to 168 kHz, the power is only 10% of the maximum value; when the resonant frequency is less than 28.5 kHz, the system efficiency drops below 50%. Clearly, the resonant frequency at which the output power peaks and the resonant frequency at which the transmission efficiency first reaches its peak are different; they cannot achieve the optimal effects of the system’s two characteristics simultaneously. The resonant frequency is mainly adjusted by the inductance and compensating capacitance, so the selection of a suitable combination of inductance and compensating capacitance is necessary during the design process.

3.1.2. Impact of Input Frequency

The system’s two transmission characteristics are superior when the system is in resonance. When the system’s input frequency does not match the resonant frequency, the two transmission characteristics indicators are:

(26)$P_{out}=\frac{{\left(2\pi fM\right)}^2{U}_S^2{R}_L}{{\left[\left(R_p+i{X}_p\right)\left(R_s+i{X}_s+R_L\right)+{\left(2\pi fM\right)}^2\right]}^2}$

(27)$\eta =\frac{(2\pi fM{)}^2{R}_L}{(R_s+i{X}_s+R_L)\left[(R_p+i{X}_p)(R_s+i{X}_s+R_L)+(2\pi fM{)}^2\right]}$

where $X_p=2\pi f{L}_p-1/2\pi f{C}_p$, $X_s=2\pi f{L}_s-1/2\pi f{C}_s$.

When the system operates in a resonant state with an input frequency of 100 kHz, the impact of input frequency on system characteristics, as illustrated in Figure 7, shows that the transmission performance achieves its maximum at the resonance point, with a rapid decline on either side of this point. When the input frequency is less than 87.3 kHz or greater than 122.9 kHz, the output power drops to less than half of its maximum value; when the input frequency is less than 87.3 kHz or greater than 129.4 kHz, the efficiency falls below 50%.

When the system’s input frequency does not match the resonant frequency, detuning occurs, which can be categorized into three scenarios: single-end detuning and both-ends detuning [29].

When the transmitting end is detuned but the receiving end is in resonance, i.e., $2\pi f{L}_s-1/2\pi f{C}_s=0$, the system’s transmission characteristics are:

(28)$P_{out1}=\frac{{\left(2\pi fM\right)}^2{U}_S^2{R}_L}{{\left[\left(R_p+i{X}_p\right)\left(R_s+R_L\right)+{\left(2\pi fM\right)}^2\right]}^2}$

(29)${\eta }_1=\frac{(2\pi fM{)}^2{R}_L}{(R_s+R_L)\left[\right(R_p+i{X}_p\left)\right(R_s+R_L)+(2\pi fM{)}^2]}$

When the receiving end is detuned but the transmitting end is in resonance, i.e., $2\pi f{L}_p-1/2\pi f{C}_p=0$, the system’s transmission characteristics are:

(30)$P_{out2}=\frac{{\left(2\pi fM\right)}^2{U}_S^2{R}_L}{{\left[R_p\left(R_s+i{X}_{s+}{R}_L\right)+{\left(2\pi fM\right)}^2\right]}^2}$

(31)${\eta }_2=\frac{(2\pi fM{)}^2{R}_L}{R_p(R_s+i{X}_s+R_L{)}^2+(2\pi fM{)}^2(R_s+i{X}_s+R_L)}$

Based on the analysis above, the relationship between the transmission efficiency of the MCR-WPT system in four states is $\eta >{\eta }_1>{\eta }_2>{\eta }_S$, indicating that the system’s transmission efficiency is highest when both the transmitting and receiving ends operate at the resonant frequency. At this time, the impedance at both the transmitting and receiving ends is minimized, and the current is maximized. Similarly, the output power reaches its peak when both ends of the circuit are in a resonant state.

3.2. Circuit Parameter Characteristics

3.2.1. Study on the Impact of Inductance

The coil inductance stores the magnetic field energy of the wireless power transfer system, and the two coils transfer energy through the time-varying magnetic field. A change in inductance values not only alters the design parameters of the transmitting or receiving end but also significantly affects the magnetic field in the transmission medium.

According to the coupling coefficient definition $k=M\sqrt{L_p{L}_s}$, the relationship between inductance and transmission characteristics can be obtained as follows:

(32)$\eta =\frac{{\omega }^2{k}^2{L}_p{L}_s{R}_L}{R_p(R_s+R_L)+{\omega }^2{k}^2{L}_p{L}_s}\cdot \frac{1}{R_s+R_L}$

(33)$P_{out}=\frac{{\omega }^2{k}^2{U}_s^2{L}_p{L}_s{R}_L}{{\left(R_p(R_s+R_L)+{\omega }^2{k}^2{L}_p{L}_s\right)}^2}$

With $k=0.32$, $R_{\mathrm{L}}=16 \mathsf{\Omega }$, coil equivalent internal resistance $R_p=R_s=0.55 \mathsf{\Omega }$, and resonant frequency $f=100 \mathrm{k}\mathrm{H}\mathrm{z}$, the impact of inductance on system characteristics is illustrated in Figure 8. When the resonant frequency is fixed, the transmission efficiency increases as the inductance of both coils increases, while the output power first increases and then decreases as the inductance of both coils continues to increase.

The transmission characteristics at different resonant frequencies are shown in Figure 9. Under different resonant frequencies, the two transmission characteristic indicators change more significantly with the increase in resonant frequency. The transmission efficiency rises with the continuous increase in inductance and remains relatively stable after reaching a certain level; the output power initially increases rapidly with the increase in inductance, reaches a peak, and then decreases quickly.

The relationship between the receiving end current and inductance is illustrated in Figure 10. This figure shows that as the inductance of both coils continuously increases, the current flowing through the load also increases, reaches a maximum value, and then begins to decrease, with the rate of decrease being slower than the rate of increase. When the coil’s inductance is very large, both the coil’s radius and number of turns must significantly increase, resulting in loss resistance and radiation resistance of the coil. The equivalent resistance of the coil increases with the coil’s radius and number of turns, leading to increased coil losses, which is detrimental to the improvement of system efficiency.

3.2.2. Study on the Impact of Mutual Inductance

Generally, the higher the coupling degree between the transmitting and receiving coils in the MCR-WPT system, the better the system performance. As shown in Figure 11, when the mutual inductance between the two coils in the coupling mechanism is small, the transmission efficiency increases rapidly as the mutual inductance gradually increases. After reaching the maximum value, the transmitting and receiving coils are in an over-coupled state, and the transmission efficiency no longer increases significantly with the increase in mutual inductance. When the mutual inductance is less than 1.99 μH or greater than 11.57 μH, the output power drops to less than half of its maximum value; when the mutual inductance is less than 4.96 μH, the efficiency drops below 50%, and below 1.64 μH, the efficiency drops below 10%.

To investigate the effect of the two coils’ sizes on mutual inductance, let us assume both coils have 14 turns, and the distance between them is 50 mm, with the transmitting coil radius being 7.5 mm. Assuming that the radius ratio of the receiving coil to the transmitting coil is $\sigma =r_2/r_1$, the relationship between mutual inductance and radius ratio can be obtained by substituting it into the empirical formula of mutual inductance, as shown in Figure 12. From the figure, it is evident that as the radius ratio increases, the mutual inductance between the two coils also increases. When the radius ratio is greater than 1, the mutual inductance is larger; therefore, it may be appropriate to design the receiving coil radius larger than the transmitting coil radius when necessary.

3.2.3. Load Characteristics

Changes in the load at the receiving end not only result in a change in impedance at the receiving end but also affect the current at the transmitting end, which in turn impacts the system’s transmission performance, primarily reflected in the output power. During the transition from a small to a large load, the output power decreases; however, if the load is too small, the energy loss on the coil’s internal resistance becomes significant. Therefore, one of the keys to enhancing system transmission performance is selecting the optimal load.

The system’s output power is given by:

(34)$P_{out}=\frac{(\omega M{)}^2{U}_S^2{R}_L}{\left[R_p\right(R_s+R_L)+(\omega M{)}^2{]}^2}$

The result of deriving the output power with respect to the load is:

(35)$\frac{\partial P_s}{\partial R_L}=\frac{(\omega M{)}^2{U}_S^2\left((\omega M{)}^2+R_p(R_s-R_L)\right)}{\left(\right(\omega M{)}^2+R_p(R_s+R_L){)}^3}$

Obtaining the extremum of the derivative yields:

(36)$R_L=R_s+\frac{(\omega M{)}^2}{R_p}$

At $R_p=R_s=0.4 \mathsf{\Omega }$, $U_s=20 \mathrm{V}$, the relationship between output power, load, and frequency is depicted in Figure 13. The system’s output power decreases as the frequency increases; during the process where the load increases from small to large, the output power initially rises rapidly, reaching a peak before decreasing as the load continues to increase. The rate of decrease is significantly slower compared to the rate of power increase.

Based on Equation (4), the partial derivative of transmission efficiency with respect to load is obtained as:

(37)$\frac{\partial \eta }{\partial R_L}=\frac{\left(\omega M{)}^2\right[R_s(\omega M{)}^2+R_p(R_s^2-R_L^2\left)\right]}{(R_s+R_L{)}^2[(\omega M{)}^2+R_p(R_s^2+R_L^2\left)\right]}$

Therefore, the extremum is:

(38)$R_L=\sqrt{R_s^2+\frac{(\omega M{)}^2{R}_s}{R_p}}$

The relationship between the two parameters of load and input frequency with transmission efficiency is depicted in Figure 14. The efficiency increases with the input frequency but reaches a critical frequency after which the system’s transmission efficiency no longer changes significantly. During the process of continuously increasing load, the transmission efficiency rapidly increases, and after reaching a peak, it decreases slowly.

As the equivalent load value of the output end of the wireless power transfer system decreases, the likelihood of frequency splitting due to frequency shift increases, significantly reducing the output power at the resonant frequency. In practical applications, the load resistance might be a lithium battery, whose resistance value changes with different charging stages. The output power under various loads can be calculated using MATLAB.

When the mutual inductance between the transmitting and receiving coils is constant, the relationship curve between the system’s output power under different loads and the input frequency is shown as in Figure 15. At lower loads, such as 5 Ω and 10 Ω, the output power exhibits two peaks as the input frequency increases, with the resonant frequency being a point of minimum output power. This indicates the occurrence of frequency splitting. As the load resistance gradually increases, the two peaks of the output power curve corresponding to the output frequency move closer together, eventually forming a single peak. At this point, the resonant frequency is a point of maximum output power, which is also the maximum value that the system’s output power can achieve.

4. Experimental and Result Analysis

4.1. Circuit Parameter Characteristics

Taking into account the design requirements of the MCR-WPT system and the characteristics of the circuit, this paper adopts a hardware circuit schematic as shown in Figure 16, utilizing a series–series compensation method.

4.1.1. Main Circuit Design

In practical applications, the parameters of the two coils within the coupling mechanism tend to be consistent. The frequency required by the system design is 100 kHz. To ensure stable system performance when the load changes, that is, to maintain strong transmission performance even when the transmitter circuit is not operating at the resonant frequency without making the coil size too large (which would increase internal resistance and reduce transmission performance), the quality factor for both circuits is set at 2. Based on the definitions of resonant frequency and quality factor, the coil inductance at one end of the circuit is determined to be 50.92 μH, and the compensation capacitance is 49.74 nF.

The high-frequency inverter circuit at the transmitter end uses a typical asymmetrical half-bridge. According to the system’s designed power, the requirements for the MOSFETs are calculated to be $V_{DS}=208~240 \mathrm{V}$ and $I_{DS}=10.4~13.8 \mathrm{A}$. To realize this inverter circuit, a pair of complementary PWM signals is required. Two N-channel MOSFETs, model IRFP360, along with drivers, are selected to implement the circuit. The main parameters of IRFP360 are shown in Table 2.

The physical image of the high-frequency inverter module is shown in Figure 17.

4.1.2. PWM Signal Generation and Driver Circuit Design

The primary function of the microcontroller is to output control signals. In this paper, the core board used is the STM32F334C8Tx microcontroller (STMicroelectronics, Geneva, Switzerland), with its main parameters shown in Table 3.

A flowchart of the STM32F334C8 outputting PWM waves is shown in Figure 18.

The microcontroller unit’s (MCU) peripherals use an 8 MHz passive crystal oscillator as the main component of the clock circuit. An external selection jumper is provided at the Boot0 pin to choose the chip’s boot mode. Additionally, the peripherals include a reset button and an SWD debugging interface, all of which provide necessary conditions for program downloading and debugging. The ME6211 series LDO is used to supply 3.3 V power to the chip, supporting up to 6 V input and capable of powering via USB.

The PWM signal controlling the MOSFETs on the transmitter side is output by the STM32 microcontroller’s IO port. The output voltage amplitude is very low, and the peak current is also not significant, being insufficient to effectively drive the MOSFETs to ensure they conduct effectively. This necessitates the use of an additional driver chip. To meet the requirements for high current and voltage, an isolated gate driver is chosen, model IR2181S, with its main parameters shown in Table 4.

The schematic of the IR2181S driver circuit is shown in Figure 19.

According to the data sheet, this driver does not have an integrated dead-time driver, which is necessary for the asymmetrical half-bridge to perform the inverting function. Therefore, when outputting the PWM signal, it is necessary to include a dead-time in the signal.

4.1.3. Receiver Circuit Design

The receiver circuit receives the high-frequency alternating current transmitted from the transmitting end, which cannot be directly used to power the load. Therefore, a rectification circuit is needed to convert the current into direct current. To minimize current fluctuations and ensure stable operation of the load, a filtering capacitor is added before powering the load. Calculations indicate that the rated voltage for the rectifying bridge is $240~320 \mathrm{V}$, and the rated current is $7.5~10$ $\mathrm{A}$. The circuit diagram of the receiver end is shown in Figure 20. The diodes used in the rectifying bridge are SR1060 Schottky diodes in a DO-127 package. These diodes have a maximum reverse voltage of 600 V, can handle a maximum current of 10 A, and their recovery time meets the system requirements.

The physical image of the receiver end’s rectification and filtering module is shown in Figure 21.

4.2. Experimental Platform

When the two coils in the coupling mechanism are in a critical coupling state and the system input frequency matches the resonance frequency, the two coils, due to their equivalent internal resistance, act as resistive loads. Considering safety factors, low-voltage experiments are conducted. The system exhibits a similar trend in transmission characteristics in low-voltage experiments as in high-voltage tests; hence, this paper substitutes low-voltage experiments for high-power, high-voltage experiments.

In this experiment, a linear regulated power supply provides a 24 V voltage input to the inverter circuit. A 220 μF electrolytic capacitor is added after the linear regulated power supply to prevent self-excitation oscillations due to the presence of a high-frequency load in the power supply circuit, thereby smoothing voltage fluctuations and achieving stable power supply to the inverter circuit. The inverter module is connected to the transmitter coil through terminals, and one end of the rectifier and filter module is connected to the receiver coil through terminals, while the other end is connected to a load resistor. The load resistor has a resistance of 16 Ω and a maximum power of 400 W. All measured voltage values in the experiment are RMS values, including the input voltage measured in the transmitting circuit into the inverter module, and the output voltage measured in the receiver circuit after rectification and filtering. The current values measured by a multimeter in series with both circuits are also RMS values.

The calculated inductance of the coil is 50.92 μH. The actual spiral coil used has a maximum outer diameter of 150 mm and 14 turns, made from 200 strands of Litz wire, capable of carrying a maximum current of 15 A. The two coils are closely placed on a ferrite plate with an initial permeability of about 3300 μ, and their self-inductance measured value is 52.33 μH. The coil and self-inductance measurement experiment is shown in Figure 22.

The experimental circuit setup is shown in Figure 23.

From the analysis above, it is known that under constant or slightly changing transmission distances, the transmission characteristics in seawater are essentially consistent with those in air. Due to the limitations of experimental conditions, not all experiments could be conducted in the deep sea; therefore, part of this experiment was conducted in air, selecting input frequency characteristics and load characteristics experiments for illustration and underwater verification.

4.3. Input Frequency Characteristics Experiment

After determining the parameters of the two coils in the coupling mechanism, selecting a capacitor can determine the circuit’s resonance frequency. The control of the input frequency is primarily managed by the output signal of the STM32, with the output PWM wave driving the inverter circuit to convert DC to high-frequency AC. Adjusting the frequency of the PWM wave, i.e., the input frequency, to match the resonance frequency is key to efficient system transmission. With an axial distance of 50 mm between the two coils, adjusting the frequency of the PWM wave, at an input voltage of 24 V and a frequency range of 90~110 kHz, by measuring the effective values of current and voltage at both ends, the output power and transmission efficiency can be obtained, shown in Table 5.

The variation curves of the two indicators are shown in Figure 24.

From the experimental data, it is observed that the maximum output power and transmission efficiency of the MCR-WPT system do not occur at the designed 100 kHz. This deviation is due to the actual values of capacitance and inductance not matching their designed values in practical engineering applications, with the compensating capacitor valued at 50 nF and the inductance measured at 52.33 μH. One reason for this discrepancy is the presence of measurement errors, and another is the inherent distributed capacitance of the coil itself. After investigating the sources of these errors, it was found that both output power and transmission efficiency decrease as the input frequency deviates from the resonance frequency to either side, confirming the previous simulation results that the transmission performance is optimal when the system is in a resonant state.

4.4. Load Characteristics Experiment

The equivalent internal resistance of the coil, measured with a multimeter, is 0.56 Ω. According to Equation (36), the optimal load for maximizing system transmission efficiency is related to three factors: the coil’s equivalent internal resistance, the system input frequency, and the mutual inductance between the two coils, with the calculated optimal load being approximately 16 Ω. When other system influencing factors are constant, the system transmission power curve under different loads will have a maximum value point. However, in actual system construction, the real load may vary due to various factors, potentially leading to a decrease in output power. In this experiment, the system’s input frequency was kept consistent with the resonant frequency, and experiments were conducted within a resistance range of 0–100 Ω. The processed experimental data are shown in Figure 25.

From the figure above, it is observed that when the load resistance is very low, the voltage distributed across the load in the receiving end circuit is low, but the current flowing through the load is high. As the load resistance continuously increases, the voltage across the load initially rises, and the current flowing through the load decreases, but the output power increases. When the resistance value increases to around 16 Ω, the output power reaches its peak. As the load continues to increase, the current flowing through the load decreases, and the output power also drops, consistent with the simulation analysis provided earlier.

4.5. Underwater Experiment

To verify the consistency of the transmission characteristics trends in seawater and air described earlier, an experiment was conducted underwater with the two coils at different frequencies. The position of the two coils underwater is shown in Figure 26.

To ensure the accuracy of simulating a seawater environment, an appropriate concentration of sodium chloride is added to the experimental water. This experiment was conducted underwater, with the two coils placed in water and their coaxial distance being 50 mm. A linear regulated power supply provided a 24 V voltage input to the inverter circuit. The load resistance was a ripple resistor with a resistance value of 16 Ω and a maximum power of 400 W.

The data of the two transmission characteristics indicators under different input frequencies are shown in Table 6.

The variation curves of the two indicators are shown in Figure 27. By comparing with the variation curve of transmission characteristics with input frequency in Section 3.1.2, it was found that the change in both indicators is consistent, with the transmission performance achieving its maximum value at the resonance point and rapidly decreasing on either side of this point, validating the accuracy of the simulation.

The comparison of system transmission characteristics between this article and other literature is shown in Table 7. Compared with the research results in references [30,31,32], it can be found that our wireless power transmission system has achieved the expected results in underwater applications, with good transmission characteristics, which can be further popularized and promote the application of deep-sea wireless power transmission technology.

5. Conclusions

Compared to terrestrial applications, underwater wireless power transmission presents unique challenges and considerations. This includes dealing with different transmission media and environmental factors. We propose a new technology tailored for underwater environments, taking into account factors such as coil design, frequency selection, and power transmission efficiency under different conditions. Based on the analysis of the principles of the MCR-WPT system, this paper established a mathematical model, analyzed the changes in self-inductance and mutual inductance of underwater coils, and made corrections to the model. Theoretical analysis and experimental verification of its transmission characteristics were then conducted, yielding the following important conclusions:

• Initially, a mathematical model of the traditional MCR-WPT system was established. It was then analyzed to determine the differences in self-inductance and mutual inductance underwater compared to in air. When the input frequency is within the MHz range, the values in these two mediums are very similar. After considering the eddy current loss in seawater, the traditional mathematical model was corrected. The corrected underwater model shows that the trend in transmission characteristics of the underwater system is consistent with that in air, laying the foundation for future analysis of transmission characteristics using traditional model performance indicators.

• The transmission characteristics of the MCR-WPT system were analyzed. It was found that as the resonant frequency continuously increases, the system’s output power first increases and then decreases, while the transmission efficiency remains almost unchanged after reaching a certain value. The system’s transmission performance is better the closer the input frequency is to the resonant frequency. In studying the system load, the optimal load for maximizing system output power was identified, but the load that maximizes transmission efficiency is different from this optimal load.

• An experimental platform was built. Experiments on the system’s input frequency characteristics and load characteristics were conducted. Experimental data were obtained and compared with simulation analysis data, confirming the validity of the results obtained in the previous study. In the experiment examining input frequency characteristics, the system’s output power reached a maximum of 15.09 W at an input frequency of 96 kHz, and the transmission efficiency reached a peak of 89.35% at an input frequency of 98 kHz. During the load characteristic experiment, the output power achieved a peak value of 14 W when the load was approximately 16 Ω. Both simulation data and experimental data show that the condition for achieving maximum output power is when the system operates at the resonant frequency and both ends of the circuit are matched. To optimize the system’s transmission characteristics, appropriate selection of various parameters in the circuit is necessary. Our research results can help us better select parameters to achieve high transmission efficiency, provide theoretical basis and technical support for the development of resonant underwater wireless power transmission systems, address the unique challenges faced by underwater wireless energy transmission, and promote its widespread adoption in practical applications, promoting exploration and development in the deep-sea field.

Author Contributions

Conceptualization, Q.H. and Y.Q.; methodology, Y.Q.; software, Z.L. and M.Z.; validation, Q.H., J.H. and M.Z.; formal analysis, Y.Q.; investigation, Y.Q., Z.L. and J.H.; resources, Q.H.; data curation, Y.Q.; writing—original draft preparation, Y.Q.; writing—review and editing, Y.Q., Z.L. and Q.H.; visualization, Y.O.; supervision, Q.H.; project administration, Q.H.; funding acquisition, Q.H. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data presented in this study are available on request from the corresponding author.

Acknowledgments

The authors would like to thank the anonymous reviewers for their constructive suggestions, which comprehensively improved the quality of the paper. All individuals have consented to the acknowledgement.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. MCR-WPT system transmission structural diagram.

Figure 2. MCR-WPT system equivalent circuit model.

Figure 3. Self-inductance of the coil under different media.

Figure 4. Magnetic field distribution in air (left) and seawater (right).

Figure 5. Variation curves of mutual inductance with [Forumla omitted. See PDF.] under different conductivities.

Figure 6. Transmission characteristics variation curve with resonant frequency.

Figure 7. Transmission characteristics variation curve with input frequency.

Figure 8. The influence of inductance on transmission characteristics.

Figure 9. Impact of inductance on transmission characteristics at different resonant frequencies.

Figure 10. Influence curve of inductance on the current of the receiving circuit.

Figure 11. The impact of mutual inductance and resonant frequency on transmission performance.

Figure 12. The relationship between mutual inductance and radius ratio.

Figure 13. The relationship between output power, load, and input frequency.

Figure 14. The impact of load and input frequency on transmission efficiency.

Figure 15. The variation curve of output power with input frequency under different loads.

Figure 16. Schematic of the MCR-WPT system’s hardware.

Figure 17. Inverter module.

Figure 18. Flowchart of STM32F334C8 outputting PWM waves.

Figure 19. Driver circuit schematic.

Figure 20. Receiver circuit diagram.

Figure 21. Rectification and filtering module.

Figure 22. Coil and self-inductance measurement experiment.

Figure 23. Transmission characteristics verification experiment.

Figure 24. Relationship between transmission characteristics and input frequency at an axial distance of 5 cm, an input voltage of 24 V, and a frequency range of 90~110 kHz.

Figure 25. The relationship between output power and load under critical coupling.

Figure 26. Coil underwater experiment.

Figure 27. Relationship between underwater transmission characteristics and input frequency.

Table 1

Main parameters of the system.

Parameter	Symbol	Unit	Parameter	Symbol	Unit
Input frequency	$f_{input}$	Hz	Angular frequency	$\omega$	$\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$
Resonant frequency	$f_{resonance}$	Hz	Load resistance	$R_{\mathrm{L}}$	$\mathsf{\Omega }$
Transmission end coil resistance	$R_{\mathrm{S}}$	$\mathsf{\Omega }$	Receiver coil resistance	$R_{\mathrm{D}}$	$\mathsf{\Omega }$
Mutual inductance	$M$	$\mathrm{H}$	Supply voltage	$U_S$	$\mathrm{V}$
Output power	$P_{\mathrm{o}}$	$\mathrm{W}$	Transmission efficiency	$\eta$	%

Table 2

The main parameters of IRFP360.

Parameter	Symbol	Value
Rated voltage	$V_{DS}$	400 V
Rated current	$I_{DS}$	23 A
Rated gate voltage	$V_{GS}$	20 V

Table 3

Main parameters of the STM32F334C8 microcontroller.

Component of STM32F334C8	Performance
Core	Up to 72 MHz main frequency
Memory	64 Kbytes
SRAM	12 Kbytes, with parity check function
Timer	217 ps high-resolution, capable of self-compensation for drift caused by power supply and temperature
ADC	12-bit, 5 Msps
Comparator	Shutdown only requires 26 ns
Operational amplifier	5 types of built-in gain, accuracy up to 1%

Table 4

Main parameters of IR2181S.

Parameter	Symbol	Value
High-Side maximum floating voltage	$V_S$	600 V
Gate drive supply voltage	$V_{CC}$	10~20 V
Logic ground	$V_{SS}$	−5~5 V
Logic input voltage	$V_{IN}$	Vss~Vss+4
Turn-On delay	$t_{on}$	180 ns
Turn-Off delay	$t_{off}$	220 ns

Table 5

Input frequency characteristics experimental data.

Input Frequency (kHz)	Output Power (W)	Transmission Efficiency
90	11.55	68.70%
92	13.69	78.67%
94	14.79	85.40%
96	15.09	88.25%
98	14.88	89.35%
100	14.07	88.89%
102	13.22	86.44%
104	12.19	84.95%
106	11.12	78.55%
108	10.09	73.63%
110	8.95	68.21%

Table 6

Underwater input frequency characteristics experimental data.

Input Frequency (kHz)	Output Power (W)	Transmission Efficiency
92	13.59	78.73%
96	15.03	88.12%
100	14.06	88.76%
104	12.14	84.81%
108	10.01	73.47%

Table 7

The comparison of system transmission characteristics.

	Frequency	Outer Diameter of Coil	Transmission Distance	Transmission Power	Transmission Efficiency
This article	100 kHz	15 cm	5 cm	14.26 W	88.76%
Reference [30]	98 kHz	9 cm	4.5 cm	14.4 W	74.6%
Reference [31]	1.8 MHz	7 cm	7 cm	9 W	75%
Reference [32]	6.78 MHz	5.2 cm	18.9 cm	7.82 W	83%

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